I'm getting the impression that there may be problems with rental cars in Europe. Insurance, fine print, places that insurance won't cover. We would really like to drive because we like to get off the beaten path. Drove once before in France, Germany, and Czech Rep. Scary situation not having a sticker in Czech Rep. Other than that, no problem. Any advice?
To rent a car in one country and drop in another one is very expensive. All destinations you named are easy to get to by train. So if you want to drive around countryside rent a car and return in the same country. That way you will also have a sticker for the country.
We plan to take the car back to Budapest because two of us are staying for another week. With no car. Still can't figure out which district to stay in for that last week. Was planning on District 7 but read somewhere that it was "a little run down". I wonder if that means it's not a district for two ladies to be walking around in the evening?
Driving on this trip should be very similar to the other European driving you have done. Just tell the car rental company the list of countries where you plan to drive.
Most of Budapest still requires restoration. But each morning they sweep up the crumbling plaster from the sidewalks and create as clean a city as any in Europe. District VII in part is the old Jewish District and WWII deportation ghetto. You can taste the history in the 7th, of course, since it is also the home of so many Ruin Pubs you can taste the beer too.