No not unless the manner of obtaining the SSN was illegal.

No not by this alone if support is sufficient.

Best to check USCIS processing times for guidance.

If you are not currently living together you may need a waiver of the joint filing requirement. Consult with an attorney for guidance.

Retain counsel for the relative to assist.

Yes this appears possible and it is best to work with an attorney for assistance with AOS.

Domestic travel with valid ID should be sufficient.

You may apply for an extension while you are visiting during the validity period of I94.

You may submit written correspondence to USCIS to inform them of the status change and upgrade request.

Yes this shouldn’t be an issue for domestic travel.