I don't know if your rental agency informed you about the vignette needed for Austria or if they said it was okay to drive your rental in Slovakia or Czech Republic.
You may have everything covered. But it would be a good idea to take a look at Gemut.com and to download and read the free What You Should Know About Renting a Car in Europe in 2018. It covers a lot of your questions.
Although it's not perfect, Rome2rio does a decent job of comparing ways to get from one place to another in terms of cost and time. For example, they say it takes 3 hrs 40 min to drive from Stuttgart to Salzburg, about 3 hours from Salzburg to Vienna, you get the idea. With stopping along the way, or maybe taking backroads, it will likely take longer. With your very short timeframe, knowing how much time you'll spend on the road is important.
We love road trips and did an Alps trip the last 2 weeks of June in 2011. We picked up the car in Stuttgart, drove through southern Germany, Austria, northern Italy, Switzerland and France, returning the car in Lörrach, Germany just over the border from Basel.
We stayed in 7 locations along the way, none of which was exceptionally touristy. Since the trip was about the driving and the scenery, we weren't particularly concerned about our 3 single night stays. Those gave us extra nights in other locations.
I did make all the reservations ahead of time. All our lodgings were in small hotels or guesthouses. All had parking for free or for a fee at the place or nearby. However, the lodging parking was very limited, so I made advance reservations for that, too.
Based on our experience, I wouldn't leave where I was going to sleep or park up to chance, even in early May. And since we were snowed in at our Austrian hotel and had to wait for the plows to clear the roads a bit during that late June Alps trip, you need to be aware of potential weather issues.
Keeping in mind that they are averages, a good place for checking is Weather2travel.com. You can switch between Metric and Imperial measurements. Be sure to scroll down your results page to see lots more detail, especially the links under "More about..." Here's a link for Cesky Krumlov.
What to do with your bags while looking around depends on what the train station has available. Some may have lockers or a place to leave luggage. Some may have nothing at the station at all, not even a toilet! This is another reason to do a bit more formal planning if you decide not to drive.