US is massive and road trips are boring because you're just in the middle of nowhere for the vast majority of it. But I grew up in the US, so maybe it's less boring for a foreigner.

The entirety of the US is extremely safe. You can probably find trouble in extremely specific neighborhoods of large cities but you'd never find your way to those places as a tourist.

Roads to avoid is a funny question. Country is huge. My general advice is don't drive at night outside of cities. There are deer everywhere and they will total your car, land you in the hospital, and end your trip. I had it happen to me once and now I plan my road trips to never travel at night. Plus it's dark and you can't see anything and you're missing the point of the road trip at that point.

Driving state to state isn't a problem. There's rarely a stop at state borders and when there is they're only there to regulate commercial trucking.

My advice is to pick a few cities and fly between them. Pick cities you can get around in without renting a car. You're not missing much by taking an 7 hour flight instead of a 5 day drive.

It's better to pick a region (or if in the West, maybe a state) and just visit that. Many foreigners don't realize how big the US really is. Doing cross-country roadtrips is a great way to suck up time doing nothing but driving.

Do something like visit New England for a month and stay in a new state each week.