The only border crossings you'd really notice (barring Hungary battening down the hatches again) are the ones to/from Croatia. You could find yourself simply facing a sign with a reminder of local speed limits, and a language change.

It is quite different. A lot of the areas you're planning on visiting are a lot more densely populated that somewhere like Utah, and with significant freight and holiday traffic. Imagine driving around Southern California, or the NE Corridor to get a better picture, plus cities without any freeway/feeder roads, and some areas with single-carriageway roads only. Driving requires more concentration, too. Summer holiday traffic, for example if you're in Italy around Ferragosto, can give you multiple hour delays on even relatively short journeys. Again, you need some detailed research, although it doesn't always get it right. For example, France regularly publishes expected peak traffic flows for holidays, but I've been stuck in a 2h-long traffic jam (that I ended up escaping from to take country roads) that wasn't predicted (while listening to a radio station discussing why it wasn't predicted).

OTOH, go off the beaten track, and you'll find some beautiful, tranquil spots.

Rentalcars.com is a decent search engine - one of many. Lots of people on the car hire boards here recommend autoslash as well. Do be wary of insurance requirements (CDW is generally mandatory, however you cover it) and the country restrictions (that will likely be documented in the small print by a specific rental company). In terms of companies, I personally find the majors in much of a muchness, including some of their discount brands. So, Hertz/Thrifty, Avis/Budget, Europcar, Sixt, Enterprise (much less of a low-cost operation in Europe than in the US). Beware of some of the loss-leader operations that rely on hidden fuel/insurance upsells and dubious charges to make their money. There are decent local rental companies, of course, but do your research - especially in tourist areas.