Ideas for 2 week trip

I suggest that your husband spends some time on Google Maps and Viamichelin to get a better understanding of driving times. For example, a drive to Davos, close to the Swiss/Austria border, is about 6 1/2 hours. Bolzano in northern Italy is about 5 1/2 hours. Not really a hop for a day trip, or even an overnight. Linz, Vienna, Melk and Salzburg for sure should be on your list, and hopefully would be by train. Au appears to be a tiny village, with the nearest train station in Mauthausen. So I'm afraid any sightseeing will be by car or train. Your husband will need an International Driver's Permit. I would consult closely with your hosts about your touring possibilities.

"hop over" to see Germany, Switzerland, and Italy too

Yes to Germany and sights in Austria. I am with you that Switzerland and Italy are a stretch.
I hope your hosts have Melk, Salzburg, and Cesky Krumlov on the to do list. IMO you should spend a couple of days & nights in Vienna.

Hopping over depends on your travel style. Yes, you can just "hop over" to Germany from Vienna, depends where.

If you're willing to consider taking the OeBB night train, the Night Jet from Vienna, in one night you'll be in Berlin or Hamburg by morning, likewise to Munich or Cologne. I've done the night option several times and still do this option . Vienna central station is a night train hub.

Vienna central station is a night train hub.

Unfortunately night trains have to compete with low cost air carriers. Unless you want to use the night trains just to not waste day time for traveling the costs may tip the scales for flights.

They have a list of things they want to show us, ...

If possible, could you publish the list here? So we could add our two cents in a more focused way.

I have no idea what the concept of the stay is, but I did take the time to look up where you are staying. That means staying full time? The time of year is good for the region, maybe a tad cold in the higher elevations, but generally nice in Vienna and south.

Anyway, if you do want to see some of the region then what looks to be a logical trip would be Au an der Donau to Linz and train to maybe Hallstatt, then Salzburg, then Munich then Prague, then a few stops through the Czech Republic, like Cesky Kurmlov on the way back to Au an der Donau which might require a bus or a shuttle service.

From Salzburg and Munich there are some interesting day trips. I usually go to a site like Viator to see what is being sold as day trips out of various cities. You don’t have to book the trip there but they provide a good list.

Budapest would be a good side trip as well, but maybe it only works if you are spending time in Vienna, otherwise it’s a bit of a long trip.

No doubt Wien Hbf is a hub for night trains and they do sell out. The Wien-Paris night connection reveals such demand for this connection. Prior to Dec 2015 City Night Line had the Paris - Munich night route, I took it twice. That route was abolished when DB ended CNL.

Now with OeBB bringing back a night route to Paris from Germany or Austria within a couple of years ago, that's good, very welcome news and favourable. I know nothing about the level competition from discount carriers, not one of my travel options, only that lots of people ride night trains, and one had better have a 2nd class seat reservation to avoid being bumped.

This is the list our student sent.

Cities: Vienna, Salzburg, Grein (because she has family there), Linz

Attractions: Stephen Dom, Schonbrum, Hofburg, Kunsthistorisches Museum, Naturhistorisches Museum, Mahu, Opera, Mauthausen Memorial, Greinburg, Klosterneuburg monastery, Baumgartenberg, Stift St. Florian, Stift Admont, Bad Gastein (they have an apartment there so we would likely stay a couple days here), Dachstein glacier, Gamskarkogel

She also has Grado & Venice on her list

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I have no idea what the concept of the stay is, but I did take the time to look up where you are staying. That means staying full time? The time of year is good for the region, maybe a tad cold in the higher elevations, but generally nice in Vienna and south.

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Well we have two weeks there. For the time we are doing anything within driving distance of her house in Au an der Donau, we'll stay there. If we travel further than would be a comfortable drive for a day trip (~3h) we'll plan to book lodging. We will also stay for a couple days at their apartment in Bad Gastein, so anything we do within a few hours of there we'd plan for that as our base.

Stephen Dom, Schonbrum, Hofburg, Kunsthistorisches Museum, Naturhistorisches Museum, Mahu, Opera, Klosterneuburg monastery,

This means at least three days in Vienna.

Well, there isn't a tremendous amount within 3 hours of Au an der Donau / Linz. I have passed by the area twice over the years, lovely to look at, but remote. Venice for example, by the time you stop for gas and food and make one wrong turn, is probably 8 hours. Train is no better. Technically Vienna is 2.5 hours. If it were me, and we are all different, I would plan on some local Au an der Donau area exploring for a few days upon arrival and then take off on a trip ending in Vienna (if thats where you flight home is from).

Well, there isn't a tremendous amount within 3 hours of Au an der Donau / Linz.

Except Linz itself, which is underrated mostly, there are Vienna, Cesky Krumlov, Hallstatt, Salzburg, Innsbruck, Munich, Graz reachable by train within 3 hours. Prague is 3:45 hours.

Because Linz is located more in the middle of Austria you have more choices than departing from Vienna or Salzburg.

You have your local guide. Stayed in Aschach/Donau for an overnight once. Nice small Donau promenade. Stopped at Cafe-Konditorei Schörgi in Grein, Austria on the way home once too. Just drove thru Linz with no visit, but what I do remember was the steel mill and the familiar smell from my younger days in Cleveland, OH. Maybe from Wien head towards the Waldviertell and visit Dürnstein or Melk? Munich, Innsbruck from Aschach/Donau. You could fly?

Yes, Salzburg and Vienna, neither is a day trip. Didnt think about Cesky Krumlov .... sure, thats a day trip. Hallstatt too. Linz is a great starting location to see a lot of great places. And I am certain you Austrian folks can point out some little known day trip destinations too. But I would do a road tour and not have to do the 3 hours there and then the 3 hours back ever day. Not when you are surrounded by some of the best of Europe. If I were to do a three hour day trip, it wouid have to be by train, direct train, so I could take the last train home and nap on the way.

I'd be very tempted to just do what your Austrian friends suggest and not stress about it. I wish I had friends in Europe who'd be willing to cart me around and show me the good stuff.

I'm sads now.