"Does it makes sense to skip Salzburg and stay in garmisch or mittenwald, since one can visit neuschweinstein castle from garmisch..."
No, that doesn't make a lot of sense to me.
If N'stein is a must-see, why stay in the Garmisch area? You could just stay in Füssen (near N'stein) and hike around that N'stein area instead. (Füssen, Garmisch, Berchtesgaden... All three are attractive places.)
But it would not be my recommendation to stay in Garmisch or Füssen, either for 1 night or for 6 nights. Your first plan allocates FOUR days for Salzburg (well, THREE days, w/ a day trip to B'gaden) probably because Salzburg is important to you for specific reasons. So why give up a place that important? AND you assigned TWO days to Munich, probably because you have sightseeing goals there - right?
So this idea of 6 nights in Garmisch seems to throw away all your plans except hiking in the Alps and Neuschwanstein.
And about N'stein... why is that the one "keeper?" As a day trip from Munich, the 30-minute tour will cost you nearly FIVE hours on trains and buses. How much do you know about N'stein? In the first place, you're confused about the spelling, which is -schwan- rather than -schwein-stein (Schwein means "pig" btw.) But what about the "Castle" part? Germans do not refer to it as a castle because it's really a palace, not a castle, a relatively modern building built in the last part of the 19th century that was given a fake castle exterior. Yes, it's an interesting building, but not necessarily a building that is worth the extra Euros required to get there, the high admission fee, and the loss of an entire day in Munich.
DW article on Neuschwanstein:
http://www.dw.com/en/is-neuschwanstein-all-its-cracked-up-to-be/a-17887035
CASTLES AND PALACES: Consider the real ones in/near Salzburg and in Munich...
So I think your first plan makes the most sense. But I could also see a 3/3 split between Munich and Salzburg. Just depends on your sightseeing goals there.