1. Your marriage certificate, itself, is a legal name change document. You can immediately, after marriage, start using your new legal name. For USCIS purposes, your new legal name can be any combination of the names listed on the marriage certificate. Use your new legal name on AOS paperwork.
2. Yes, you can apply for a Social Security number before marriage. Use your maiden (Pre-marriage) name for the application.
Another option is to wait until naturalization to change your name. In our case, wife has never changed her name, and it has been fine.