They are almost certainly referring to your EAD. You should have a lawyer assist you with your VAWA. You would have a much higher chance of success.

You would need to provide sufficient documentary evidence to demonstrate that the applicant would not be inadmissible pursuant to INA 212(a)(4)...

If she is not receiving the medical treatment, she should not make a misrepresentation by suggesting that she will be traveling to the U.S. for...

No, it should not affect your K-1. However, you should consider having a lawyer assist with a K-1. Statements that you make at your K-1 interview...

Crimes that render an alien deportable are set forth in INA 237(a)((2). These include aggravated felonies (237(a)(2)(A)((iii), which are further...

The use of an EAD granted based on an pending I-485 will cause you to violate your H-4 status. It will be considered unauthorized employment.

You can call the USCIS Contact Center at (800)375-5283.

CBP could ask about her. However, based upon the facts that you provided, it does not appear likely.

It is not a good idea to consciously overstay and remain unlawfully present in the USA. See INA 212(a)(9)(B). Speak to a lawyer to discuss options...

You have to research the other documents that may be available in his home country. Lawyers assist with this process frequently.