~~One post split off to its own thread.~~
Your visa will match your Passport. If you want to change your name the cheapest and easiest is to do it before the interview by getting a new passport in your married name. If your passport in already in married name, then there is nothing left to do. The visa cannot be in any other name but the one in your passport.
If you wait until after you move it will cost a good chucnk to change the green card. Sorry I can't remember the cost atm. Then the next chance will be at ROC if you enter as a CR1, or if you go for citizenship you can fill out the forms with the new name. You do not have to have an official/court name change but many go that way too.