10 days ausria - all nature type

pick up a good book Rick Steves Austria. you extremely behind, lodging may be hard to find since all the good inexpensive ones go fast. Switzerland is better for natural beauty, but I have heard western Austria is fairly close but I have not been there yet. You do not mention whether you like hiking(walking down miles of trail and then back to the start) or pulling up hopping out of a car and snapping a photo and leaving. There is a difference.

10 days, does that count arrival and departure day? If so you have 8. Your day of arrival will be useless due to jet lag. Driving the same day you land after an overseas flight is impaired driving. Land and take the train to Salzburg. then pick up a car. I would stick to trains. They are easy and fast.

I recommend Hotel Hohenstauffen in Salzburg. Its close to the train station and a pretty 10-15 minute walk along the river to the old town. In Salzburg, I particularly recommend the baroque cathedral. (I know you mentioned no churches, but this one is spectacular) It's interior is white with lovely stained glass and quite different from other cathedrals I've visited in Europe. Mozart was baptized there. I also enjoyed the funicular up to the fort for the great views and at the base of the funicular, St. Peter's cemetery. It was the inspiration for the Sound of Music cemetery and the cemetery and its setting is charming. I did the Sound of Music tour, but I'm CRAZY about the Sound of Music. They take you into the mountain/lakes area. You could take a train day trip to Hallstatt.

I wouldn't rent a car. I find train travel when it gets me where I want to go, quite relaxing. The train trip from Munich to Salzburg with the Bayern ticket is easy and very reasonable. There is also a very inexpensive train ride to the Westbahn train station in Vienna. It was under 30 euro and no advance purchase is necessary. It is a train company that competes with OBB. I'm sorry, I don't know the name of the company.
You could consider Vienna, the architecture is incredible, you could just wander the streets and enjoy. The Naschmarkt is very interesting and great for pictures. You can take a trolley to the Heurigers just out of town. The area is very pretty and you can do hikes from there. Heurigers are charming wineries that also serve food--think beer garden but wine. There are also plenty of options for visiting small towns on the Danube from Vienna--train, boat, bus, even biking. In Vienna, I highly recommend Hotel Austria, but you may need to take whatever you can get.