HRQX

The sentence on the USCIS page is incorrect and N-600K can be filed online from abroad as OmarStuck did.
 
In fact, the vast majority of online applications are submitted from abroad. The DOS regulations even state that the children easily overcome INA 214(b) concerns when the parents live outside of the US: https://fam.state.gov/fam/09fam/09fam040202.html "Naturalization under INA 322 is a permissible activity in B-2 status. You may issue a B-2 visa to an eligible foreign-born child to facilitate that child's expeditious naturalization pursuant to INA 322. The child must be under the age of 18 at the time INA 322 requirements are met. The child's intended naturalization, however, does not exempt the child from INA 214(b); the child must intend to return to a residence abroad after naturalization. A child whose parents are residing abroad will generally overcome the presumption of intended immigration, provided that the parents do not intend to resume residing in the United States, whereas a child whose parents habitually reside in the United States will not."

That shouldn't be an issue. Even in the short period when the previous Administration's public charge rule was in place there were many things that were not considered "public benefits" https://web.archive.org/web/20210303222548/https://www.uscis.gov/policy-manual/volume-8-part-g-chapter-10
Other benefits not considered public benefits in the public charge inadmissibility determination include, but are not limited to:
Any services provided under the Robert T. Stafford Disaster Relief and Emergency Assistance Act (Stafford Act);[21] 
Benefits under the Emergency Food Assistance Act (TEFAP);[22]
Child and Adult Care Food Program (CACFP);
Food Distribution Program on Indian Reservations (FDPIR);
Short-term, non-cash, in-kind emergency disaster relief;
Programs, services, or assistance (such as soup kitchens, crisis counseling and intervention, and short-term shelter) provided by local communities or through public or private nonprofit organizations;
Public health assistance for immunizations with respect to immunizable diseases and for testing and treatment of symptoms of communicable diseases whether or not such symptoms are caused by a communicable disease;
Attending public school;
Benefits though school lunch or other supplemental nutrition programs including:
Benefits through the Child Nutrition Act;[23]
Benefits from the National School Lunch Act;[24]
Summer Food Service program;
Child care related services including the Child Care and Development Block Grant Program (CCDBGP);[25]
Special Supplemental Nutrition Program for Women, Infants, and Children (WIC);[26]
Children’s Health Insurance Program (CHIP) and State Children’s Health Insurance Program (SCHIP);[27]
Health Insurance through the Affordable Care Act;
Tax Credits;
Transportation vouchers or other non-cash transportation services;
Housing assistance under the McKinney-Vento Homeless Assistance Act;[28]
Energy benefits such as the Low Income Home Energy Assistance Program (LIHEAP);[29]
Educational benefits, including, but not limited to, benefits under the Head Start Act;[30]
Student loans and home mortgage loan programs; and
Foster care and adoption benefits.
As there are multiple federal and state public benefits programs, USCIS is unable to list all programs not included within the public charge inadmissibility determination.

In general, send the following four packets (*I-765 and I-131 packets are optional but free) in the same envelope to the Chicago Lockbox:
I-130 with I-130A and copy of marriage certificate, filing fees, etc.; I-485 with I-864 and supporting documents, I-944 and supporting documents, electronic I-94 copy, filing fees, etc.; I-765 with electronic I-94 copy, etc.; and I-131 with electronic I-94 copy, etc. You'll separate each packet with rubber bands or fasteners: https://www.uscis.gov/forms/forms-information/form-filing-tips
On the I-130 packet, I-130 payment method (check, money order, or Form G-1450) must be the very first item of this packet.
On the I-485 packet, I-485 payment method (check, money order, or Form G-1450) must be the very first item of this packet.
And they already married recently so they aren't eligible for K-1.

Mail the withdrawal letter after he leaves the US.
If you follow the aforementioned advice then there won't be any overstay (I'm assuming that Form I-485 was properly filed before his I-94 expired) https://fam.state.gov/FAM/09FAM/09FAM030211.html

I-693 is not needed. https://www.uscis.gov/policy-manual/volume-8-part-b-chapter-3
K or V Nonimmigrants Applying for Adjustment [26]
K and V nonimmigrants applying for adjustment of status are not required to repeat the medical examination if the application was filed within one year of the date of the original medical examination, and:
The medical examination did not reveal a Class A medical condition; or
The applicant received a conditional waiver in conjunction with the K or V nonimmigrant visa or the change of status to V and the applicant submits evidence of compliance with the waiver terms and conditions. [27] 
If a new medical examination is required and reveals a Class A medical condition, a new waiver application will also be required. In such cases, the officer should determine whether the applicant complied with the terms and conditions of the first waiver, if applicable. Such determination should be given considerable weight in the adjudication of a subsequent waiver application. [28] 
 
Even if a new medical examination is not required, applicants must still comply with the vaccination requirements if the vaccination record was not included as part of the original medical examination report. If the vaccination report was properly completed at the time of the overseas examination, the officer may accept the vaccination assessment completed by the panel physician.
 
An applicant’s overseas medical examination report completed by a panel physician should already be in the applicant’s A-file. If it is not in the A-file, the officer should request the medical examination report through a Request for Evidence (RFE).

I assume the children don't have SSNs. Thus, I recommend submitting Form I-765. With either EAD, GC, or US passport they'll then be eligible for SSNs. If they pay the I-485 fees, then both I-765 and I-131 are free to file: https://www.uscis.gov/sites/default/files/document/forms/i-485instr-pc.pdf
In general, send the following twelve packets (*note that I-765 and I-131 packets are optional, but highly recommended) in the same envelope to the Chicago Lockbox:
I-130 with I-130A and copy of marriage certificate, filing fees, etc.; I-485 with I-864 and supporting documents, I-864A and supporting documents, I-944 and supporting documents, electronic I-94 copy, filing fees, etc.; I-765 with electronic I-94 copy, etc.; I-131 with electronic I-94 copy, etc.  
I-130 and copy of child's birth certificate, filing fees, etc.; I-485 with a letter (or old Form I-864W) stating that the child is exempt to Affidavit of Support requirement, I-944 and supporting documents, electronic I-94 copy, filing fees, etc.; I-765 with electronic I-94 copy, etc.; I-131 with electronic I-94 copy, etc.  
I-130 and copy of child's birth certificate, filing fees, etc.; I-485 with a letter (or old Form I-864W) stating that the child is exempt to Affidavit of Support requirement, I-944 and supporting documents, electronic I-94 copy, filing fees, etc.; I-765 with electronic I-94 copy, etc.; and I-131 with electronic I-94 copy, etc. They can separate each packet with rubber bands or fasteners: https://www.uscis.gov/forms/forms-information/form-filing-tips
On the I-130 packets, I-130 payment method (check, money order, or Form G-1450) must be the very first item of those packets.
On the I-485 packets, I-485 payment method (check, money order, or Form G-1450) must be the very first item of those packets.
 
If the children have received public benefits listed on Page 8 of I-944 instructions do not list those benefits on I-944 as they are exempt from that specific part of the Public Charge rule: https://www.uscis.gov/sites/default/files/document/forms/i-944instr-pc.pdf See page 10 "Children Acquiring U.S. Citizenship" row.

Did you get the new one from "Consejo Supremo Electoral"? https://travel.state.gov/content/travel/en/us-visas/Visa-Reciprocity-and-Civil-Documents-by-Country/Nicaragua.html If so, print-out the preceding link's section on Birth Certificates.

Do not apply for a (a)(6) EAD!! If you do apply for it, then it will be a futile attempt.
 
After marriage, file I-485, I-131, and I-765. The (c)(9) EAD and AP applications will be free if full I-485 fee is paid: https://www.uscis.gov/i-485
Filing Forms I-765 and I-131 with Form I-485
If you submit Form I-485 and pay the required fee, you do not have to pay an additional fee to also file Form I-765, Application for Employment Authorization, and/or Form I-131, Application for Travel Document, for advance parole. You may submit these forms together. If you choose not to file Form I-765 and/or Form I-131 with your Form I-485, then you must submit a copy of your I-797C, Notice of Action, (also known as your receipt) as evidence that you filed Form I-485.

*** Moved to Tourist Visas forum ***

As a paper document:

As a paper document:

As a paper document:

https://uk.usembassy.gov/visas/immigrant-visa-faqs/applying-for-an-immigrant-or-fiancee-visa-2/
We are traveling to the United States to marry and will return to the United Kingdom after marriage. Do I still need a fiancé(e) visa?
If you will return to your permanent residence after the marriage ceremony, you may apply for a tourist B-2 visa, or if eligible, travel visa free under the Visa Waiver Program.  At the time you apply for the visa and/or travel to the United States you will be required to show that you have a residence outside the United States that you do not intend to abandon. There is no set form that this evidence takes as it varies with each person’s circumstances.

https://uk.usembassy.gov/visas/immigrant-visa-faqs/applying-for-an-immigrant-or-fiancee-visa-2/
We are traveling to the United States to marry and will return to the United Kingdom after marriage. Do I still need a fiancé(e) visa?
If you will return to your permanent residence after the marriage ceremony, you may apply for a tourist B-2 visa, or if eligible, travel visa free under the Visa Waiver Program.  At the time you apply for the visa and/or travel to the United States you will be required to show that you have a residence outside the United States that you do not intend to abandon. There is no set form that this evidence takes as it varies with each person’s circumstances.

https://uk.usembassy.gov/visas/immigrant-visa-faqs/applying-for-an-immigrant-or-fiancee-visa-2/
We are traveling to the United States to marry and will return to the United Kingdom after marriage. Do I still need a fiancé(e) visa?
If you will return to your permanent residence after the marriage ceremony, you may apply for a tourist B-2 visa, or if eligible, travel visa free under the Visa Waiver Program.  At the time you apply for the visa and/or travel to the United States you will be required to show that you have a residence outside the United States that you do not intend to abandon. There is no set form that this evidence takes as it varies with each person’s circumstances.

https://uk.usembassy.gov/visas/immigrant-visa-faqs/applying-for-an-immigrant-or-fiancee-visa-2/
We are traveling to the United States to marry and will return to the United Kingdom after marriage. Do I still need a fiancé(e) visa?
If you will return to your permanent residence after the marriage ceremony, you may apply for a tourist B-2 visa, or if eligible, travel visa free under the Visa Waiver Program.  At the time you apply for the visa and/or travel to the United States you will be required to show that you have a residence outside the United States that you do not intend to abandon. There is no set form that this evidence takes as it varies with each person’s circumstances.

On March 8 USCIS shared the following alert: https://www.uscis.gov/i-131

On March 8 USCIS shared the following alert: https://www.uscis.gov/i-131

On March 8 USCIS shared the following alert: https://www.uscis.gov/i-131

The I-129F petition is processed by USCIS. Having an approved I-129F doesn't necessarily result in DOS continuing the K-3 process. DOS is the one that issues visas: https://travel.state.gov/content/travel/en/us-visas/immigrate/family-immigration/nonimmigrant-visa-for-a-spouse-k-3.html
Important Notice:
When both petitions have been approved by USCIS and sent to the NVC or when USCIS approves the I-130 before the I-129F, the availability of, as well as the need for, a nonimmigrant K-3 visa ends. If the NVC receives both an approved I-130 petition and an approved I-129F petition:
The nonimmigrant K-3 visa case will be administratively closed. The application process explained below will not be available to the foreign-citizen spouse and cannot be used. The NVC will contact the U.S. citizen sponsor and foreign-citizen spouse, with instructions for processing the IR-1 (or CR-1) immigrant visa. For more information on the immigrant visa process, review the Immigrant Visa for a Spouse webpage.

On the front just the middle initial, but on the back of the EAD it shows the full middle name on my card.
 
For example, below the middle name is "Void"

If she is aged 14-78 then it's correct: https://www.uscis.gov/feecalculator But if she's aged 79 or older then it's a bit less.
Yes
When did the Civil Surgeon sign the medical? Also, a reminder that the medical cannot be older than 60 days when you resubmit it:
https://www.uscis.gov/policy-manual/volume-8-part-b-chapter-4 "The civil surgeon signs Form I-693 no more than 60 days before the applicant files the underlying benefit application with USCIS" https://www.uscis.gov/i-693 "Medical examinations signed by the civil surgeon Oct. 1, 2021, or later must use the 09/13/21 edition. We will continue to accept prior editions for medical examinations signed by the civil surgeon before Oct. 1, 2021. You can find the edition date at the bottom of the page on the form and instructions."  
Also, the filing address may have changed.
you live in:
Alaska California Commonwealth of the Northern Mariana Islands Connecticut Delaware District of Columbia Florida Georgia Guam Hawaii Illinois Indiana Iowa Kansas Maine Maryland Massachusetts Michigan Minnesota Missouri Nebraska New Hampshire New Jersey New York North Carolina North Dakota Ohio Oregon Pennsylvania Puerto Rico Rhode Island South Carolina South Dakota Texas U.S. Virgin Islands Vermont Virginia Washington West Virginia Wisconsin Then file at:
USCIS Chicago Lockbox
U.S. Postal Service (USPS):
USCIS
Attn: AOS
P.O. Box 805887
Chicago, IL 60680
FedEx, UPS, and DHL deliveries:
USCIS
Attn: AOS (Box 805887)
131 South Dearborn - 3rd Floor
Chicago, IL 60603-5517
 
you live in:
Alabama Arizona Arkansas Colorado Idaho Kentucky Louisiana Mississippi Montana Nevada New Mexico Oklahoma Tennessee Utah Wyoming Then file at:
USCIS Phoenix Lockbox
U.S. Postal Service (USPS):
USCIS
Attn: AOS
P.O. Box 20500
Phoenix, AZ 85036-0500
FedEx, UPS, and DHL deliveries:
USCIS
Attn: AOS (Box 20500)
1820 E. Skyharbor Circle S
Suite 100
Phoenix, AZ 85034-4850
Last Reviewed/Updated: 11/15/2021

I-130 and I-864 are filled-out by the US citizen, and the rest of the forms are filled-out by the immigrant. In general, send the following four packets (*note that I-765 and I-131 packets are optional, but highly recommended) in the same envelope to the Chicago Lockbox:
I-130 with copy of your naturalization certificate, copy of your birth certificate, filing fees, etc.; I-485 with I-864 and supporting documents, electronic I-94 copy, filing fees, etc.; I-765 with electronic I-94 copy, etc.; and I-131 with electronic I-94 copy, etc. She'll separate each packet with rubber bands or fasteners: https://www.uscis.gov/forms/filing-guidance/form-filing-tips
On the I-130 packet, I-130 payment method (check, money order, or Form G-1450) must be the very first item of this packet.
On the I-485 packet, I-485 payment method (check, money order, or Form G-1450) must be the very first item of this packet.
 
She'll eventually have to go to a doctor that has Civil Surgeon designation from USCIS. https://www.uscis.gov/policy-manual/volume-8-part-b-chapter-4
Timing of the Submission of the Medical Examination Report
Applicants may submit the Form I-693 medical examination report to USCIS:
Concurrently with the immigration benefit application; or
At any time after filing the immigration benefit application but before USCIS finalizes adjudication of that application. If not submitted simultaneously with the immigration benefit application, applicants may bring the medical examination report to an interview or wait until USCIS issues an RFE requesting the medical examination report.

If OP is aged 21 or older then OP's mother would become an Immediate Relative upon OP taking the naturalization oath: https://www.uscis.gov/policy-manual/volume-7-part-b-chapter-8
Immediate Relatives
Certain adjustment bars do not apply to an immediate relative, including the spouse or child (unmarried and under 21 years old) of a U.S. citizen, and the parent of a U.S. citizen older than 21. [2] 
An adjustment applicant applying as an immediate relative may be eligible to adjust status even if:
The applicant is now employed or has ever been employed in the United States without authorization;
The applicant is not in lawful immigration status on the date he or she files the adjustment application;
The applicant has ever failed to continuously maintain a lawful status since entry into the United States;
The applicant was last admitted to Guam or the Commonwealth of the Northern Mariana Islands (CNMI) as a visitor under the Guam or CNMI Visa Waiver Program and is not a Canadian citizen;
The applicant was last admitted to the United States as a nonimmigrant visitor without a visa under the Visa Waiver Program; or
The applicant has ever violated the terms of his or her nonimmigrant status.
No. OP cannot currently (nor previously) file I-130 for parent because OP is a LPR.
 
But after OP naturalizes then the I-130 can be filed.
Check her I-94 record: https://i94.cbp.dhs.gov/I94/#/recent-search Her B-2 status most likely ended way before 2021 assuming an Extension of Status (EOS) was not filed and approved.