GWagon

As of right now, your petition for your brother will take about 15 years.  If your mother becomes a US citizen, yes.

My answer doesn’t change.  Until your brother marries he is not married, and so there is no reason to foreclose an optimization. 
 
 I also don’t know what happens if he marries and then the marriage terminates before his visa number becomes available. The category says: “unmarried son”. It doesn’t say “never married son”.  

Because he is going to marry then your Mother needs to be a USC to file for him, so 6 years probably from now.
 
You can file now and she can file later.

Both you and your mother should each file I-130 for your brother, because the back logs are measured in decades.   Statistically there is a good chance one of you passes away before your brother’s visa number becomes available.  
 
If he marries after his mother becomes a U.S. citizen, that’s fine (but your mother will have to inform USCIS to modify the I-130).  Otherwise because there is no immigration visa category  for LPRs petitioning married sons, your mother will have to file a new I-130.  

You can Apply any time and yes you can do it yourself and save attorney money. The process is fairly straightforward.

Congrats

Is it still sealed? If so, you can still use that I-693. https://www.uscis.gov/sites/default/files/document/policy-manual-updates/20211209-I-693FilingReqt.pdf
 
https://www.uscis.gov/i-693 "We will continue to accept prior editions for medical examinations signed by the civil surgeon before Oct. 1, 2021."

Sure looks like it!  That is a stroke of luck.  I'd say resubmit asap, and you should be fine.

Yup, they just don't do that.  They don't have the staff to call people who make errors, unfortunately.

Isn't the I-485 supposed to be $1225?

I'd use the next day delivery so it gets delivered tomorrow.

In order to use the existing medical it must arrive at the Chicago Lockbox on or before 11/20. See 8 C.F.R. 103.2(a)(7)(i): "USCIS will consider a benefit request received and will record the receipt date as of the actual date of receipt at the location designated for filing such benefit request whether electronically or in paper format."

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.

Yes, you will print all the forms, sign on the I-130 and I-864, your mother will sign on I-485, I-765, and I-131. If you are submitting I-693, a certified civil surgeon has to sign off on it and seal it in an envelope (do NOT open it).
 
No, this will go to the Chicago Lockbox, yes all in one envelope:
 
Via USPS:
USCIS
P.O. Box 805887
Chicago, IL 60680-4120
 
Via FedEx, UPS, Etc:
USCIS
Attn: FBAS
131 South Dearborn - 3rd Floor
Chicago, IL 60603-5517

No, I-485 cannot be filed online which throws a wrench into the whole plan.

Yay for same day oath. That is one less thing to worry about.

I will have to ask her and make that decision. Here in NJ, they offer same day interview/oath.

Be prepared for all possible outcomes.  Look at my timeline, 2+ years from initial naturalization petition to oath ceremony.  
 
Good luck 

Also check the time it takes Between now and your N400 interview to Oath add that to the time it will take to AOS. 
 
Most importantly have you asked your mother if she wants to stay stuck in the US (AP is not taking 3 to 5 months to get it's more like 8+ months). Will she have medical insurance incase of emergency, how about family back home? Will she want to fly back if an emergency happens there? Make sure before she stays long enough to get a 10 ban that she is 100% sure this is what she wants to do. 

Basically what @nastra30 said. You will file I-130, I-485, I-765, I-131, and I-864 and all supporting documents all at the same time in the same envelope. I-693 (vaccination record) you can file either together with the whole packet or she can send it in later when she gets an RFE or bring to the interview if USCIS decides to interview her. You can file as soon as you get your naturalization certificate.

Yes; you can send I-130(Petition), I-485(AOS), I-765(EAD), I-131(AP) all at the same time in one packet, right after you naturalize and you have naturalization certificate in hand.

I'm not sure advance parole wipes away the ban for overstay parents but I could be wrong. Others can chime in.

I will add my 3 cents that as a part of AOS application she can and will file I-765 for interim employment authorization (EAD), and I-131 for interim travel authorization (advance parole). Both are free if filed with I-485 or while I-485 is pending. Both take roughly 6 months to be issued and if filed together the advance parole just gets attached to the EAD card as an annotation.
 
So it's not like she'll be with nothing throughout the entire process while waiting for her green card. Roughly 6 months in she'll get her EAD and be able to work, get a social security number, get a driver's license, travel internationally (be mindful of covid restrictions, AP alone does not bypass those), etc.
 
She should NOT travel abroad until she has the EAD/AP in hand. Doing so now would get her banned for re-entry and others already established that she would not be eligible for a waiver.

In that case AOS is the best option. If she leaves the country she will incur a ban and will have to serve that ban before been let back in.