Have the university file for you as a researcher or professor.

A work permit based on adjustment of status does allow you to work wherever you want.

If CIS approved your I-485 you should already have your green card.

There is no appeal. Perhaps she could file a lawsuit for a declaratory judgement.

There may not be any rule against it but based on common sense, I would not recommend it.

Appeal the FOIA decision. Clearly they have records relating to you.

They can ask whatever they want, but CIS will not revoke your green card due to your employer failing to keep their agreement. See a labor lawyer...

They will no doubt contact NVC to see if it was cancelled.

You are out of status. Upon approval, you were required to join your second H1b petitioner. Perhaps CIS will allow you a 60 day grace period so...

If there is a question on the application that asks you about that, you should answer it.