what to expect when moving after filing green card through marriage application?

[Coleman21 6]

I am married to an American citizen and I am currently on F-1 visa, but we are going to apply for green card this month or next month, depending on when my parents in law get their income tax return. However, due to financial reason, we are thinking of moving to my husband's parents' house who is our joint sponsor in another state. (our aparentment rent is very expensive here and we cannot afford my tuition from next semester because my parents stopped funding me. I am thinking of taking a leave of absence in August.)

My questions are: 1. Is it ok to move/change address before receiving NOA? (We want to apply before moving because we have a lot of proof that we are living here, but if we apply from my husband's parents' house we won't have lease or bills with our names.)

2. Even if we file application before moving, will we still have to establish additional proof after moving that we are still living in the same house, in our case my husband's parents' house?

3. General question: I still use my maiden last name and hasn't legally changed my last name after marriage. I hear that these days it is becoming common, but would that be a reason for the immigration to suspect our marriage?