35 weeks is a good time to do it, you don't want too early because god forbid someone rear ends you, you'd need to get a whole new car seat before baby even comes!
No. We did both of ours at 35w. My son was born at 36w and my daughter 38w.
No, I had my car seat installed at 30 weeks I think. Mostly to get it out of the house and so I didn't have to worry about it. I'm currently 37 weeks also.
I'd have done it weeks ago, you just never know when baby will be ready.
The hospital won't let you leave if you don't have a seat installed anyways.
One less thing to worry about later on!