Yes, you can use your green card to be admitted to the U.S. as long as it is not expired.

Perhaps her work will not be unauthorized. But her presence in the U.S. would be unauthorized.

Hard to say. Perhaps it is a mistake.

The AR-11 is required to notify USCIS where you live, and you can also use it to notify USCIS where you want your mail sent.

If you have a work permit you can work as many jobs as you wish.

If he is in the US in legal status, he can file for adjustment if you petition for him as a permanent resident. If the i-730 is approved instead,...

You cannot get another naturalization certificate based on changing your name after you got the certificate.

You do not need to prove you maintained lawful, nonimmigrant status, to be eligible for adjustment of status based on marriage to a US citizen.

You can check the processing times on the USCIS website.

You file at your local court for a name change.