Yes if previously sponsored, she should be indicated.

Contact the embassy directly to inquire.

Contact USCIS directly by writing to inform them of status change.

You must establish your domicile in the US to proceed.

No you should be able to travel with validity issued GC and passport.

This is possible but the previous marriage will be scrutinized.

Wait for USCIS to request more evidence through an RFE.

Your suggested move is not advisable.

Write a letter directly to the asylum office processing your case.

No this should not detrimentally impact your immigration status.