Raising Premiums in a No-fault State?

I would like some advice on what to do. I live in Utah, which from what I understand is a no-fault state. On November 19, I got rear-ended on the interstate. It was a pretty straightforward accident, the other driver, Adam (not his real name) was apologetic and it was 100% his fault. No one was injured. A report was made with the Utah Highway Patrol.

The next morning, I got 2 quotes from different body shops, and they both said it would be minimum $4,000, but there is likely unseen damage, so the total is likely more than that, not including my rental car. I have USAA, Adam has Progressive. I filed a claim with Progressive the next day, but the update from them is still that they're waiting on Adam's statement. Adam has not been responding to my texts.

My car is still drivable and the trunk still opens and closes, so I'm not on a time crunch. My questions are:

1. How do I force Progressive to cover the damages/Adam to give them a statement?
2. If I go through USAA, even though it's a no-fault state, would I still have to pay my $500 deductible and have my premiums raised? That wouldn't be fair to me.
3. When my car is being fixed, how do I get insurance to cover my rental car? Or will the insurance company walk me through that?