Lemonslice

Imo, she has been lucky.  The Carrier Information Guide specifically states: "Expired Conditional Resident Card, Form I-551 accompanied by Original Form I-797, Notice of Action indicating the card validity is extended".  The guide is not a USCIS document.  
 
"The U.S. Customs and Border Protection (CBP) Carrier Information Guide serves as a reference aid for travel industry personnel. It outlines the required documents that apply to persons entering or departing the United States. Carrier personnel are urged to familiarize themselves with all sections of the publication and reference it in their examination of travel documents."
My point is that, regardless of what USCIS says, there is a possibility she will be denied boarding.   Good luck on the rest of your immigration journey.  Next action for your spouse is either an I-751 interview or N-400 submission. 

Advance parole cannot be filed before adjustment of status. Expedite request is never guaranteed. There's no official timeline for expedited requests.
 
You can see estimated processing times for AP here https://egov.uscis.gov/processing-times/
 

Careful with what USCIS customers support says. They're not employees for the airline.
USCIS can say that you can use the copy but if the airline says you can't board with the copy then you won't be able to.
Period.

Do you have physical watermarked extension letter? The first issue you may face is airline not accepting the printout and denying boarding. Some were lucky to board with copy but it's more of an exception than rule.

Did you receive the paper copy?
Remember that is risky for her to travel with the downloaded version instead of the physical copy.
If she tries to use the online version, is a gamble. She can be denied boarding and lose the money spent on the tickets.
Regardless, let us know how it goes. Good luck!

Tier 1 reps tell all sort of nonsence all the time, unfortunately. And the worst thing is: you cannot hold them accountable. There's so many people who followed bad advice from those folks only to find VJ was more factually accurate source of information. Travelling with original extension letter is challenging from time to time, let alone a copy.

I have seen cases where a copy worked.  I have seen cases where it didn't work.  However, the official Carrier information guide says the original extension letter must accompany the expired Green Card:
 
Carrier Information Guide- English (cbp.gov)
 

Only original. Copy is a gamble. Airline staff may deny boarding with a copy.

Correct, but it is still important to use a computer and Adobe products.  Also important to input in order, so that following fields are activated or deactivated.
 

Like others have said, she needs to start a new citizenship application. In the meantime, besides ESL classes, I would do full immersion at home: speaking English only together at all times, watching TV in English with English subtitles on, reading books in English, limiting time spent with her Spanish-speaking friends, etc. It will be beneficial for her way beyond the interview. Not speaking fluent English (doesn't have to be perfect but a conversational level) will be limiting for her at work and in daily life. She might as well use this opportunity to put in the time and effort to improve her language skills.

To vote (as an answer) is a little too simple… work with your wife on the why… why does she want to vote? If she had given a more full answer first time the follow up question most likely wouldn’t have been asked… this is why going to citizenship classes will be advantageous.it will help her with giving and practicing great answers. 

The superfast are outliers, you should expect the processing time to be somewhat longer than that.  

Your timeline in your account is still incomplete. Please complete it.
Did you include a copy of the receipt for the I-130 with your AOS package?
What do you mean "restart"? I thought you had already sent the AOS package...

The superfast are outliers, you should expect the processing time to be somewhat longer than that.  

The superfast are outliers, you should expect the processing time to be somewhat longer than that.  

The superfast are outliers, you should expect the processing time to be somewhat longer than that.  

USCIS forms are not Apple or "device" friendly.

CR1 is quicker and WAY cheaper as you'll skip adjustment of status.
 
You can even skip removal of conditions if by the time of entry to the US, you've been married for two years.
 
Forget the K1, not an option anymore -except in very few cases where for some reason the couple cannot get married-.

Don’t do any of this, not a good plan! 

This is such a bad idea.  CBP are very well trained at detecting things like this. 
What if they call your employer to verify your employment?
What if they see you have a 2 week trip booked and they limit your trip to 2 weeks?  Then you must return home with no job?
 
No one on here is going to condone or support your plan.  I suggest you plan a trip for 2 weeks, return to your home county, continue to work until you move to the U.S. on your K-1 visa.

Good luck

It seems you are suggesting trying to deceive CBP when you enter by showing them proof of a job you no longer have and showing them a return ticket you plan to change.  This is not a good plan. 

Wait, you're planning to lie about visiting for only 2 weeks then turn around and stay for 5-6 months, lying that you have a job (which you're going to quit before you depart) and you're wondering what? What could possibly go wrong ... 
 
 

Please complete your timeline. We can't tell you what to do without knowing if you're inside our outside the US!

If everyone are US Citizens then there is no Immigration issue.

What about the children's grandparents (your parents)? Do they have the physical presence requirements to transmit citizenship to your kids?