1. Yes, my SSN card doesn't have any restriction wording on it.
2. I renewed my DL with the certificate of naturalization so it's good for 8 years.
3. Yes, I'll have to update my DL with my new address. I can do this without proof of citizenship.
4. I don't think so - as a LPR I wasn't able to get a DL until the green card was received.