Please critique my draft itinerary

Budapest
You could stop in Gyor and Pannonhalma on the way to Budapest and break up the drive.
If you like wine country you could visit Etyek, or a know a good wine guide that could do a day trip with you to some small family owned vineyeards.

If you do either, then that extra days goes there.

It's a lot of driving. Just note:
May 8th is a State holiday in CZ but it should not affect tourist activities.
Czechia Driving - Make sure your hire car is insured to travel to Hungary and that it is registered as having a CZ motorway vignette (online now, not a sticker)
Hungary Driving - If you cannot do it in advance then you'll need to stop at the border to register online for the Hungary motorway vignette.
Austria Driving - You'll need to stop at the Hungary/Austria border for the Austrian motorway sticker.
Plzen - aim right for the centre i.e. the cathedral, and there are lots of parking spaces as you'll probably be there only a couple of hours. Otherwise use regular car parks. Note that you'll need to find parking where you can pay by card in case you may not have sufficient coins.
Karlovy Vary - you might need only a half-day in the centre so consider other towns within 30-40mins drive like Marianske Lazne, Frantiskovy Lazne, Cheb, Loket etc.

May 11 (Sun) pick up rental car, drive to Mukilov to visit wine
region. Overnight Mukilov

This, if done well, this one activity is a good reason for a car, or you could hire a guide that knows where and what. The rest of the trip can be done without a ca, but the car is nice luxury. I bet when all added up (rental, gas tolls, parking) the car cost $1500 to $2000. Which might be about the same price as using trains and private transfers. The car will serve no purpose in any of the destinations.

Thank you so much for these valid points. Having grown up in Wyoming, where we sometimes drove 4 hours for a high school football game, we don’t really think of it as a lot of driving, but I hadn’t completed my research on rental cars and driving through multiple countries so I will look more closely at that. Dealing with luggage (carry-on size only!), getting to and from train stations, and timing were concerns that led me to renting a car. I will map out the trip using trains to consider that.

I read an excellent article about biking around Mukilov so I will look to see if we can travel by bikes for activities there.

I truly appreciate your input. Thank you again for sharing your expertise.

Be sure the car-rental agency you plan to use is open on Sunday. Many are not.

The idea of only two days in Budapest makes me sad, but your preference for smaller towns is noted.

Dont know about Brno, but it seems to be a larger, more industrialized city. We enjoyed Mikulov in September, and visited several wineries in the area. Znojmo, and Vrbice are big wine producing towns near there.

Cesky Kriumlov is a gem. Take the guided raft trip .that makes a circle around the town center.

getting to and from train stations, and timing were concerns that led me to renting a car. I will map out the trip using trains to consider that.

I get it, I am from Texas. Never rode a subway in the United States, never rode a train in the US (except at DisneyLand). So both were sort of strange experienes at first. I live over here now and ride one or the other or both, daily. Its a snap once you get past the unknown. All of the places on your list will be full of English speakers, the signage will as often as not, be in English as well as the local language. Oddly enough, I suspect there is a greater percentage of English competent people in Budapest than in Germany. Something to do with the native tounge being unique in the world and its either learn English or be isolated. And its required in school these days.

If you were starting in Budapest I would take you over to the train station and give you a tour. Even hop on one and show you how everything is laid out. Pretty simple.

Okay, for the train tickets, schedules and costs, you go to the national train company in the country that the train origniates. You can google for that. Dont go to any thrid party sites. Thats for planning and budgeting. Before you actually buy a ticket check the same train with the national company of the destination country. Might be cheaper for the same train. If is just a few dollars less, best to use the originating train company so if there is a problem they are there to help you. If its 25 euro cheaper, wellllllllllll nothing ever goes wrong and 25 euro is real money.

Brno has a very attractive, non-touristy (on my 2018 visit) historic center, but the city is fairly large. It's reasonable to prefer a smaller place as a contrast to Budapest and Prague.