He should make sure he has receipt for payment for the green card production fees, and with that he should follow up with USCIS.

You may wish to consult someone who practices employment law regarding your legal recourse, but generally no, your employer is never required to...

Yes, since his most recent entry was not recorded, he does have to register under the new registration procedure.

Yes, you can travel domestically in your current status. Yes, you can do that, the answer to your is yes, you can travel domestically.

The answer is yes, the H4 can enter the U.S. on the H4 even if the H1b is in the process of transferring to a new employer.

I agree that travel may well be risky. We do not know what travel bans could come up, as they did just four years ago during Covid. Could be...

They could detain her and she will have to prove to an immigration judge that she did not abandon her status.

USCIS considers new certificates as sign of fraud. Check the reciprocity tables for what certificate is needed, at travel.state.gov

Yes, coming into court if one is not in legal status is indeed risky that one could subject oneself to removal.

If you can only name one, then name the primary one. If it is your step mother, then you indicate that she is your step mother.