I covered much of that ground this year. You will have a great time. I depended solely on public transportation, so my ability to visit rural areas was limited, and I have never been to Belarus.
Rather than Eger (which is in eastern Hungary), I think you may mean Győr. I liked both Győr and Sopron, but Győr has the larger historic district, plus a number of attractive small museums that I enjoyed. In addition, the Pannonhalma Abbey is within striking distance of Győr.
I think you might save some money by arranging to be picked up by your car and driver in eastern Hungary rather than Budapest, but I have no experience using such services. I'll bet James E can make some very helpful suggestions. Debrecen is a place of some size, much closer to Ukraine that Budapest is. Or perhaps--depending on exactly where you want to go in Ukraine--Kosice, Slovakia, might be useful.
Now, Ukraine: I went to the three major tourist destinations (L'viv, Kyiv and Odesa) plus three smaller cities that are less often visited by foreign tourists: Chernivtsi, Ivano-Frankivsk (just an overnight stop) and Uzhhorod. (You will see spelling variants on all of these names.) I was surprised to find only one tourist office in all the places I visited. It was in L'viv. So print maps out before you go for any city/town you might pass through, and do some Googling to see what is there. Two of the three hotels I used in the less touristy cities did not have a local map they could give me.
There are lots of young people who speak English quite well in the more touristy Ukrainian cities; in the other areas, such folks are scarcer. Everyone tried to be very helpful, but I think it could be a bit discombobulating for someone who has no experience with a language written in the Cyrillic alphabet. It would be helpful if your driver spoke Ukrainian as well as English, but I don't know how many such people you'll find in Hungary.
It seems to be recommended that tourists drink bottled water in Ukraine, as many of the locals do. I think you'll prefer the still water. Much of the fizzy stuff seems extremely salty. If you order lemonade in a restaurant, it will typically be made (in my experience) with that salty, fizzy water. The fizz was refreshing; the saltiness was not a plus for me.
The roads in the Ukrainian countryside are sometimes in extremely poor repair. I took a bus through the Carpathians, and the driver wove back and forth along several sections of the road in an attempt to avoid the worst of the potholes. So you may need more time than you think to move from place to place. I suspect there might be a problem taking a rental car from Hungary (or Slovakia) into Ukraine. Even if there isn't, the driving could well be a lot more challenging than you are expecting, and you should certainly be sure of good insurance coverage.
When traveling around Ukraine you can expect to encounter a lot of pit toilets. I found them very clean. But they were still pit toilets.
The more I read about Poland during my trip prep, the more places I wanted to go, and it's a large country requiring a good bit of time to move from place to place. I ended up spending five weeks there without covering all that I wanted to see. Ultimately, I had to call a halt or I would never have set foot in Czechia. (I had already eliminated Slovakia from my itinerary.) I can't really argue with the Big Three of Krakow, Warsaw and Gdansk, but there are so many other interesting places. You may want to concentrate on just part of the country, figuring you can make another trip to see the rest.