Naturalization happens if you apply to become a citizen, while citizenship happens if you have a birth parent who is a citizen of the U.S.
You need to reside continuously in the U.S. for five years as a lawful permanent resident before applying for citizenship.
Dual citizenship is allowed in the U.S., but your country of origin may not allow it.
You can be out of the U.S. for more than six months before you could possibly lose citizenship.
You can extend a student visa by requesting an extension through your Designated School Official (DSO) well before your program ends.