Travel to Austria research

I need some detailed info about travelling to Austria from Raleigh / Washington DC.

VISA requirements for a non US citizen. Docs required, where to get the VISA made if I am based in Raleigh, NC, etc

Best time to visit to avoid the expensive touristy season, yet not go there in the worst of times.

Best places to visit if I am interested in mountains, nature walks, hikes of upto 6 miles, adventure sports, chill outdoors...

Hotels to stay in - prefer breakfast included, room for three people family, and a room rate range of 100-150$ per day.

How to travel within Austria? Trains, cars etc. Rented car driving requirements. Where to get train tickets? How to make payments in Austria - carry cash? withdraw cash there?

Where to eat? Prefer vegetarian. Non-veg ok. Mediterranean, Indian, Thai, will be great...

What flights to take so that we do a layover in Austria for a week on the way to India? Or on the way back from India - final destination in India could