Feasibility, multi-day travel through Austria.

Check what the international drop fee is to rent a car in Austria and drop it in Germany. Your car will have to be transported back to Austria before it can be rented again. You might consider going directly to Neuschwanstein from the Grossglockner journey, then driving to Innsbruck and drop the car and travel to Munich by train. You don't want a car in Munich anyway.

I take it your desire is to drive the Grossglockner and see the castle? Because
there is (as always) good stuff that you're passing by. And it's quite a bit of driving
for what is really the few hours you'll spend seeing that stuff.

Possible alternative - train to Salzburg, spend a night or two there, pick up car, do
the drive to Grossglockner/Innsbruck. This adds time in Salzburg that you don't
have in your plan, but I think I'd rather spend time in Salzburg over Innsbruck.
And I think you save on having the rental car for fewer days.

Totally agree with Sam that you should avoid dropping the car off in a different
country, esp when the trains are as plentiful as they are.

Make sure you account for the Austria toll vignette.

A car rented in Austria should have a toll vignette included, but worth checking to make sure. You will also need to have an International Drivers Permit.

I think you underestimate the distances you are describing. The train trip from Prague to Villach is 10-12 hours door to door. Then a full day of driving the next day to Innsbruck. You will need to suck up and pay drop off fees in Germany if you stick to your plan. To be honest, there are better ways to see mountains and castles without totally exhausting yourself. I would recommend you consider taking the train to Salzburg, visiting that area and then continuing on by train to Munich. Munich is really pretty boring, in my opinion, so better spend your time in Salzburg and the surrounding alps.

"Leave Villach, drive over Grossglockner, stop at Krimml Falls, arrive in Innsbruck." Unless you don't plan to stop, I think you need more time. Something has to give. I would drive secondary roads. Weather can be fickle too.

Instead going by train from Prague to Villach you should consider a flight from Prague to Vienna and further to Klagenfurt.

Thank you all for the suggestions and comments, a lot of great points and info here. I think the plan is going to need some heavy modification and probably need to re-prioritize what we want to see.