Yes this shouldn’t be a problem with validly issued GC.

You should be able to depart with valid visa without issue.

Yes you may search for work but will need the EAD to work.

You may only travel outside the US with advance parole.

Submit form I90 to rectify the error.

This is marriage fraud and completely contrary to immigration law.

No this is not an option for recovery status.

Best to speak with USCIS directly about your inquiries.

You should have proof of payment of taxes from last tax year for affidavit of support purposes.

The answer appears to be yes and you should bring all criminal records pertaining to incident.