recommendations for Austrian Alps and Vienna?

Staying in Innsbruck is not a bad idea. It's a major transit hub with access to the mountains. Salzburg is more scenic but slightly less convenient for mountain access.

July is peak season and you really need to book your hotels ASAP.

Seefeld is a pleasant town with a nice lake for swimming and lots of alpine walking. Also lots of pensions offering accommodation. There is train service to Innsbruck, Munich, Salzberg and Vienna. Another alpine town is Mutters in the mountains above Innsbruck.

Other than in the US, most universities in Austria don't have campuses, i.e., their buildings are scattered over a larger area of a town when having become bigger over the time. Do you know to which building of the University of Vienna you want to have easy access to?

Given the excellent public transport in Vienna usually the location of your accommodation does not matter much. But in this particular case it may be more important, because between mid July and end of August 2026 there will be tram track maintenance works, i.e., five tram lines terminating at the main building of the University of Vienna will be not in operation.

The summer holidays for schools (9 weeks) and universities (14 weeks) are used for maintenance of streets and public transport every year, because there is considerably less traffic and demand given that a lot of people leave the city for vacation.

Hiker's paradise is the town Seefeld in Tirol high above Innsbruck, i.e. in a plain having an elevation of 4000 ft, providing 100+ hiking routes of different length and severity.
https://www.seefeld.com/en/hiking-trails.html

You shouldn't have difficulties selecting hiking trails matching your requirements and your physical condition.

Seefeld can easily be reached by a scenic train ride of 36 minutes from Innsbruck in case you prefer to stay in the city.

When coming from Munich, I recommend taking the train route via Garmisch-Partenkirchen for going either to Seefeld or Innsbruck. The ride is longer than the usual route but much more scenic.

This is defintely a touristy place, but (not only) for a teenager it might be fun going there:
https://www.kristallwelten.swarovski.com/en
There is a shuttle bus from/to Innsbruck.

Thank you all! After looking into some of the places y'all recommended, I'm wondering if we should just stay in Garmisch-Partenkirchen. My daughter really wants to visit a castle and go on an alpine slide.

In terms of travel logistics, the pro is that its closer to Munich so we could head straight there rather than overnighting in Munich. The con is that its a long train ride to Vienna.

Any thoughts?

We stayed in both Berchtesgaden, Germany (not far from Salzburg) and Garmisch-Partenkirchen, for several nights in each area, with our teen daughters in summer of 2024. I would recommend either of these areas as a nice stop before heading to Vienna.
Both areas have more than enough to do between hiking and a variety of outdoor activities, nearby cities to explore, thermal baths, history, etc.

We found small apartments with kitchens and washer/dryer to rent in both towns, which helped with the budget when considering meals & laundry. We did have a rental car.
Since you'll be travelling by train, you'd want to pick a location with convenient train connections from Munich and to Vienna.

Have fun!

I will focus on the Vienna portion of your trip (but I will say that Salzburg is the most magical city with a castle and you can easily stay alongside the lovely lake Wolfgangsee in Sankt Gilgen for breathtaking views and an alpine slide as well as a mountain cable car).

As mentioned above, the University of Vienna is all over the city - different departments and schools are scattered. Alas, no campus life, no mascots or sports teams like in the US. It would be helpful to know which school/address. That said, and as also already mentioned, there is absolutely no need to be on top of the university location. You can get across Vienna in minutes with our efficient and cost effective (and easy to use) public transportation.

All of that said above, you want to try to stick to districts 1, 4, 6, 7, 8 or 9. These are compact, well located districts with great public transportation connections and good shopping/restaurants. Are you looking at a hotel or an AirBnb?

Once you have picked your accommodation, come back and let me know your choice and I can suggest restaurants, etc.

I know you didn't ask, but for activities (as the mom of a 15 year old), I love to rent motorboats from Marina Hofbauer at the Alte Donau station of the U1 subway line. I also think the Carriage Museum at Schonbrunn is underrated. My favorite museum in Vienna is the Wien Museum, which is very interactive and has English explanations - and it is free.

Maybe somewhere near the Kaisergebirge. Kitzbühel