So, I think the schedule works out something like this:
Nights of 20-22: Munich
Nights of 23-24: Innsbruck
Nights of 25-26: Salzburg
Night of 27: Hallstatt
Nights of 28-30: Vienna
Innsbruck: There are a number of Almabtrieb in Tyrol on Sept 24. This is a tradition where cows are brought down from their summer high Alp grazing pastures to their valley homes. The cows are usually dressed up for the event with flowers, and there is always some sort of festival in the town. My personal opinion is drop anything else you have planned and go to one of these -- rural Tyrol/Alp culture at its finest. BUT... bad weather can always postpone it to the next weekend, so have a backup plan. You can search for YouTube videos to show you the towns and previous Almabtrieb in them. https://www.tyrol.com/things-to-do/culture/tradition/cattle-drives
Salzburg: One of my favorite small cities. I would think about getting there reasonably early to explore the city on your partial day there. For nature stuff around Salzburg, I would suggest a day trip to Königssee or Werfen Ice Cave on your full day in Salzburg. Königssee is a gorgeous lake. There is a nice hike at the far end of the lake to Röthbach Waterfall, though I've only seen it in Spring -- not sure what it is like in fall. Take the electric boat across the river to the Salet stop, walk to Obersee, hike along the right side of the lake (rocky in places -- be careful if wet), hike a relatively gentle ascent to an alpine field surrounded by granite peaks, and then head to the waterfall on the other side of the meadow. You can take a path up the side of the waterfall, but you would need some pretty good hiking equipment for that. I love that hike (though, again, have done it twice in May). At Werfen, there is a castle, but there is also an ice cave that is pretty cool. Head to the Werfen train station, walk about 10 minutes to a shuttle that will take you up to the cave entrance. Then head up to the entrance, catch a cable car, hike a little further up, and then go through the cave. I thought it was cool, too.
Königssee: https://www.berchtesgaden.de/en/nature/lake-koenigssee (Röthbach Waterfall pictures [and view of the ascent] start at around 1:11 on the video)
Ice Cave: https://www.eisriesenwelt.at/en/