No this shouldn’t be a problem for later purposes.

You should indicate all addresses, even temporary.

A cosignor is likely necessary for your case.

This will not result in a visa cancellation.

Apply for a reentry permit to travel.

Consult with an attorney for guidance.

Check USCIS webpage for extension of TPS.

A misrepresentation such as this is serious with consequences for GC. Consult with an attorney for guidance.

Depends on what this is in reference to. Consult with an attorney for guidance.

It depends on your status in the US and entry requirements to Mexico.