"From 1 November to 15 April you may only drive in winter conditions, such as icy roads or roads covered in snow or slush, if you have winter tyres fitted. If you are caught without winter tyres in Austria in such conditions, you will be fined €35."
Also, you need to buy a vignette at the border, about 20 Euros.
High airport tax on cars at MUC. I saved big bucks renting in town. Spend firs 1-2 nights in Munich-- not only saves money on car but you won't sleep well on the flight, you will have jet lag, and you are going to be driving in a very deferent place. Taking a day to recover.
Don't just familiarize yourself with International road signs, memorize them. Make every body going memorize them because the driver will appreciate help.
EU drivers follow much closer than Americans. This is unnerving. but understandable once you see how little room there is pass on EU roads. On multi-lane highways, always keep right except to pass. 4 lane Expressways have a fast and slow lane, much less speed variation than in USA. 6 lane expressways have slow, moderate, and fast lanes. Pick one and go with the flow. Road signs are very systematic. This is a good lesson USA could learn.
Don't be discouraged, driving is the best way to go, and its easy outside the big cities.
Neuschwanstein Castle is unfinished inside and as noted, not worth the effort. On the other had, viewed from above (Marienbruck bridge) or below (the parking lot, a restaurant across the way, it is spectacular. There are tables in the restaurant H. Lisl und Jaegerhaus the overlook both castles. The interior to see is the nearby Weiss Kirch.
Mad King Ludwig had two more castles, Linderhoff and The Rezidenze in Munich. Both have better interior sights than Neuschwanstein & Hohenschwangau Castles.
Keep Prague at the end. It is the best thing you will see, and by then you will be over jet lag and can fully enjoy it. Munich, on the other hand, benefits from visiting while brain dead.
Your itinerary is too tight. If Day 2- Berchtesgaden (Eagles Nest, Lake Konigsee, Salzbergwerk) has bad weather, flip it with Salzburg. Organize things as good weather (scenery) days and bad weather (towns, castles, museums) days and play it by ear.