3- Budapest to Vienna (train or bus?)
Train. Purchasing in advance will save you about 30 euro last time I looked (29 euro vs 60 euro more or less). BUT, there is no flexibility this way. Someone may have more current numbers. Or try the Seat 61 website. For my money, unless there were six of us I would buy tickets at the station so I could be flexible.
4- Vienna to Cesky Krumlov (bus?)
CK Shuttle or Bean Shuttle. Yes, I would book in advance to get a seat on the departure I wanted. Cesky Krumlov is a wonderful town, but I would travel down late one morning and leave early the next afternoon. In that time you can see a lot of it and your time will be more necessary in either Prague or Vienna or Budapest depending on your interests. I would say add 2 days to Budapest and 1 day to Gyor & the Archabbey at Pannonhalma, but that's just my preference.
5- Cesky Krumlov to Prague (train or bus ?)
CK Shuttle or Bean Shuttle. Yes, I would book in advance to get a seat on the departure I wanted.
You didn’t ask about Rome to Budapest, so at the risk of being chastised for offering something not asked, you can reach Budapest non-stop for under $75 from Bologna, Rome, Milan and Bari (either Wizzair or Ryanair).
Also, not asked; the theaters close in mid-June and all of July. So if that sort of thing interests you ......