Travel outside US on F1 after marrying US citizen but before applying for green card?

[Gesualdo]

My fiancee and I are thinking about getting married soon, but I would like to go back home to Europe for Christmas. If we get married but don't start the adjustment of status process, will I have problems reentering the US using my student visa as I always do? Will the immigration officer even know that I got married? Since I've had my F1 visa, I've never had any problems: they just check my papers and ask me for my address and maybe what I study.

If I decided to apply for my green card right away instead, how easy would it be to get advance parole so I could fly to Europe to visit my family and attend conferences and master classes related to my major?

Thanks