treppenwitz

DOS not USCIS
Not necessarily; the majority of families in similar scenarios have options. As shown in the videos, that living situation is nothing new. Both my parents were deported and I crossed daily to school since the age of 7. Same with my older siblings, distant cousins, some of my classmates and best friends. ~57% of the students in the Nogales, AZ public school district live where I grew up in Nogales, Mexico. Most of us are now working good careers in the US as electricians, entrepeneurs, engineers, hedge fund mangers, HVAC techs, park rangers, professors, etc.

DOS not USCIS
Not necessarily; the majority of families in similar scenarios have options. As shown in the videos, that living situation is nothing new. Both my parents were deported and I crossed daily to school since the age of 7. Same with my older siblings, distant cousins, some of my classmates and best friends. ~57% of the students in the Nogales, AZ public school district live where I grew up in Nogales, Mexico. Most of us are now working good careers in the US as electricians, entrepeneurs, engineers, hedge fund mangers, HVAC techs, park rangers, professors, etc.

They can still live together... in Mexico. It's not totally unthinkable for the family; the immigrant already lives with 2 of the USC children in Mexico:
The family can also move to a Mexican border town and still attend U.S. schools:
 

You can try to walk-in on an earlier date:
Sometimes they allow walk-ins.

OP can contact the DOJ's Immigrant and Employee Rights Section: https://www.justice.gov/crt/immigrant-and-employee-rights-section

No bar for overstay exists here. It sounds like AOS was filed while still in the original 90 days so there's no overstay. Even then, no bar exists for those who depart the US with valid AP. See Matter of Arrabally.
Edit: Actually, Matter of Arabally might not quite be the right case here as that's about what is considered a departure. If the I-485 itself is withdrawn/denied, then a departure does seem to occur. But the point remains that no overstay bar should exist.

You have "authorized stay" allowed by the US Attorney General.

Anytime before the visa expires is fine.

Received date on noa1

Facebook or YouTube immigration service?

Your fiance can leave North America* right now. Then try to travel to the US with ESTA (for a short trip) in late May.
*Not just the US but also Mexico, Canada, and the Caribbean.

It'll be $535 more expensive since an I-130 will also be required: https://www.nolo.com/legal-encyclopedia/if-the-i-94-k-1-visa-expired-can-i-still-adjust-status.html

USCIS also accepts credit cards: https://www.uscis.gov/news/news-releases/uscis-expands-credit-card-payment-option-fees

Apply for her Viet passport is first step or she is not going anywhere.

It's free. Send the I-131 with a photocopy of your I-485 NOA1 to the Chicago Lockbox (if the receipt notice starts with MSC): https://www.uscis.gov/i-131-addresses "Applicants with a pending Form I-485"
Odds are it will be a separate I-512 document:

It's free. Send the I-131 with a photocopy of your I-485 NOA1 to the Chicago Lockbox (if the receipt notice starts with MSC): https://www.uscis.gov/i-131-addresses "Applicants with a pending Form I-485"
Odds are it will be a separate I-512 document:

An RFE can occur at any time prior to adjudication (approval./denial), even after the interview (i.e. missing documents after the interview).

He can file again right now.

AOS depends on local field office. A handful take only a few months. Most are taking around a year. Some are taking 18-24 months.
 
A typical CR-1/IR-1 case from filing to interview is typically taking ~12-16 months, depending on various factors like embassy/consulate involved. Some are lucky to only take ~10 months. Some are waiting 18+ months.
The 7-10 months is about right for the I-130 alone. But that's just step 1 in an immigrant visa case. Then it's off to NVC for a few months of processing. Then waiting for an interview at the embassy once the case is closed at NVC, which usually adds ~1-3 months.
 
Your case is a little different in that your I-130 was already filed (although it's not approved either)...I'm not sure if it will go to the back of the line or not.

He doesn't have to wait the 30 days to refile (the 30 days is if he wants to APPEAL the first decision)..he can refile for the N400 at any time.

That only applies until the I-94 expires. After that date, you have an authorized stay since you filed AOS, but not technically a legal status.
The "Date Admitted To" will be the expiration of the I-94.
Once somebody gains employment authorization, they will show up in SAVE. Or if they obtain AP and re-enter using it,. they will be admitted as a parolee with an indefinite period of stay. Nothing in that document covers an AOS applicant without those benefits.

This was recently changed.
While you are admitted D/S, once your SEVIS record is terminated, you are considered out of status. However, previously this did not accrue unlawful presence unless USCIS or an IJ determined that you were out of status. With the change in 2018, unlawful presence starts accruing the moment the SEVIS record is terminated.
See https://www.uscis.gov/sites/default/files/USCIS/Laws/Memoranda/2018/2018-08-09-PM-602-1060.1-Accrual-of-Unlawful-Presence-and-F-J-and-M-Nonimmigrants.pdf
 
Note that any overstay is not an issue when you have AP. See Matter of Arrabally.

This is not common, but has happened a number of times in the past. USCIS works very slowly. Regardless if they decided to reschedule an interview or not, or what you told somebody on the phone, they likely haven't noticed that she left the US prior to AP being issued yet. Nobody who has actually looked at the case or that you've spoken to could verify or act upon that information. The IO at the interview can.
 
As noted, she would likely be denied AOS at the interview. Technically, it is possible to show that there was such a severe emergency that even emergency AP was not possible to be obtained prior to departure, and they have some discretion there. But I absolutely would not rely on that. Expect a denial.
 
You can refile. No new I-130 would be needed - just a new I-485/I-864/I-765/I-131.
 
Technically she would be a parolee if she uses AP. Intent isn't an issue here.
Had she try to use a B-2 or other similar visa instead of AP, then that's a much bigger issue.
 
Counterpoint: If AP was issued in error, it is not valid at all.
But I doubt it will be an issue in this case. No way to know until she tries.

No. All applications that meet the following criteria are sent to the Chicago Lockbox; https://www.uscis.gov/i-485-addresses:
They initially send it to the National Benefits Center. Then they send it to your local Field Office. If you move during the process, then fill out a change of address within 10 days: https://egov.uscis.gov/coa/displayCOAInitForm.do

Just select a few chats. It's recommended to just include about 6 pages.