If you have not used EAD before, your status is H-1b even if you have EAD. Normally, it takes about three months to get the H-1 transfer approved.
Even I will hold H1B in the new university, I can still use AP to go back, not H1b. Please correct me if I am wrong.
-Yes, but I still suggest you to get an H-1 visa if you want your spouse to get an H-4 visa to enter U.S. later after you get married.