agrabs

If they are outside of the US, your only option is consular processing. Having them enter on a tourist visa with the intent to adjust status is fraud. 
 

Your sister should try to apply for an emergency visitor visa , She can prepare by getting a letter w diagnosis from father’s doctor. 

I am sorry your dad is suffering. 
 
By the way, BIG letters are for emphasis only and you were getting excellent info only..zero attitude.
 

Healthcare costs for a new immigrant should be a major consideration before an immigration journey even starts.  Even senior immigrants must have been in the US for 5 years before being eligible for Medicare. 

She faces several obstacles:
1.  Daughter is going to be a US citizen soon.
2.  Indigent.  She cannot pay for her own trip.
3.  No strong ties to her country.
4.  Her activities could be considered work...which is not allowed on a B2 visa.
In other words, she is a prime suspect to stay and adjust status instead of following the legal route to live in the US.
She can certainly apply for a B2, but I would not get my hopes up.  I notice you used the term "live with us..."  I would use "visit us".  Good luck.

Some days..some posts, really do need an @Timona 😂 presence.

OP, don't be tempted to just hide the facts, even if your wife used fake credentials (fake name, and random SSN) to conceal her real identity.
Don't take the chance of underestimating the capabilities of USCIS and Homeland Security, because it may lead to serious repercussions. 

Visitors buy Travel Insurance, Medical Expenses is part of the cover, arguably the most important section.
 
I am assuming she has a 10 year multi entry visa bit not all are issued that way. I have seen them issued with a trip time restriction, but that is advisory to the PoE.

As the other poster said, what you're planning is immigration fraud. 

If you are already planning on adjusting status, it is immigration fraud. 

Then why did you start this thread saying you had no idea why the ESTA was denied?  Obviously the overstay is the reason.  The date CBP requires a visitor to leave is not just a suggestion.

It could have been 4 minutes, look at the VWP qualifications, she was never eligible to use the VWP.
 
Loads of people on here have visited multiple times whilst their K1 was in process.
 
Now why she was refused ESTA I do not know, and you probably never will, just be grateful as otherwise she could have had a trip and been bounced by the PoE, a long way home.

There have to be consequences for working without authorization. Otherwise, people would not bother to get work authorization.
 
* If ICE determines you have worked without authorization you can be detained and ICE can issue you an NTA. Your I-485 case with USCIS will not move forward until the court dismisses your removal case
 
* If in the course of a normal CBP (or internal stop by BP,, which is arm of CBP) inspection your unauthorized work is discovered, you can be held for ICE. Then see above.
 
* You must disclose your unauthorized work to USCIS. This can have long term consequences.  CBSA can deny NEXUS.
 
* The fact your unauthorized work history can be shared with CBSA suggests future difficulties visiting Canada unless you are a Canadian citizen.
 
* I don’t know if there are consequences for security clearances

Ask your attorney.  The Terms of Service dictate discussion here. 

USCIS must forgive. ICE and CBP do not have to forgive. 

This is the exact reason why people have trouble obtaining B2 visas, because of people like this who think that they are above the law and cheat the system.. Honestly, I hope they get caught and I hope that one day adjustment of status from a B2 is abolished.

You are missing the point.  Getting married or even rushing to get married to get a CR-1 just because it has more features and more convienent is foolish for many people.
Granted, for yourself, or for many other people, depending on their stage of life, it may make good sense to decide on it based on the features.
 
But for many others, they should decide based on whether they are really are ready to be married and then seperated for 2 years, or be engaged for 2 years before the K-1 is approved.  Its a very emotional issue, that can vary greaty depending on the couple and circumstances.
So pushing a cr-1 on many peopke and keep telling them its better may be 100% correct or 100% wrong.
 

If you applied in June 2022, it really hasn't been that long and you definitely are not outside of normal processing time. Some people don't get these documents for closer to a year nowadays. Not being able to work or travel are things people need to strongly consider when they try to jump the line by adjusting within the US instead of doing consular processing in their own country.

Reading between the lines
 
Your husband came to the US on a F1 student visa 13 years ago and has now graduated, used up OPT etc
 
You married and filed to adjust his status but supplied very little supporting documentation.
 
Have you been interviewed?
 
Most people certainly on here do not have children together.
 
I would suggest a competent lawyer, not a difficult case but appears you need help putting your case together.

They’re not looking for a traditional wedding. They’re looking for joint bank accounts, joint credit cards, joint leases/utilities, joint tax returns, joint health insurance, joint car titles/car insurance, etc.

Find a new insurance company that doesn't have that requirement. There are hundreds of insurance companies in this country, best thing to do is go online and use Google, call around and start comparing prices and coverage. Google is your friend. I literally just changed my car insurance and homeowners insurance last Thursday, if you can bundle insurance (sign up for more than one policy ie:car and homeowners or renters) you will save lots of money. I found all the info I needed on Google. I switched from Liberty Mutual to Progressive and saved a few thousands dollars combined between my two policies for the same, exact coverage. It took me a few days and hours and hours on the phone to find the right fit.

You aren't the first to ask this question.  I am simply giving you FACTS! Don't confuse direct facts with rudeness.  

I haven’t said I’m gonna do it. I was just asking if it was something possible. You don’t need to be so rude. 

Then, why even do it????  Your decision....Your risk..... YOUR CONSEQUENCES.   Just be prepared to be denied at the K-1 interview......then you would have to start over with a spousal visa. 
Being "too married for a K-1 and not married enough for a spousal visa"  is a real thing.
 

But, that's what you signed on for.......right?  Surely you must have thought about these things when you petitioned him at his advanced age?
 
SMH.  Sorry, but I think it is irresponsible and cruel to bring a very elderly person here and not provide him with basic health insurance.  

Who petitioned him?  That person is on the hook for supporting him, not the US taxpayers.