Yes.
Depends.
Is your new spouse from your country of origin?
No
I entered as an F-1 student, married a U.S. citizen, adjusted status to LPR, divorced, naturalized, K-1ed a fiancee, married her, she adjusted status to GC, she RoCed, and then she naturalized.
She has entered the U.S. as a U.S. citizen once so far, and yesterday she had her first encounter with an internal border patrol checkpoint as a U.S. citizen. Each encounter was without incident.
I am Canadian, my wife is Burmese. We did not know each other before I
* entered the U.S. as an F-1
* received my GC