Just got back from Budapest-Vienna-Prague-Cesky Krumlov-Salzburg-Munich.
Salzburg was my least favourite, so, given your time constraints, my suggestion is that you don't try to add Salzburg if you want to include Budapest in your trip.
Budapest and Prague were my favourite cities. Nymphenburg Palace in Munich was one of my favourite palaces. If you have only a couple of days in Munich, stay in Munich and see that as well as the Residenz. (I didn't go to the Ludwig castles, though, so I can't really compare.)
All those places were hot most days in July, and all had some days that were cooler/hotter than others. Our hottest day was one of our Budapest days (+37). It was bearable because 1) we had a place with air conditioning, and 2) Budapest has the baths, so there is someplace to go to get wet.
I do think your schedule is a bit ambitious. We had 20 nights plus our days/nights flying to and from Europe, and our days were full. Mind you, we do like to take our time to get to know a place a little and explore (e.g. walking the moat path in Prague, going to 3 different baths in Budapest, etc.).
I think someone else may have mentioned this, but geographically, it would make more sense to stop in Vienna in between Prague and Budapest. Consider doing something like Munich-Prague-Vienna-Budapest, or vice-versa, to minimize your travel time and maximize your sightseeing time.
Since you asked, given your time constraints, I do suggest you consider dropping one city. It would make sense to go to Munich-Salzburg-Vienna-Prague (or vice versa), for instance. Or leave out Munich and Salzburg, and just go to Prague, Vienna, and Budapest.