Vienna-Innsbruk

You don't say it you are arriving in Vienna after a long international flight. If so, that first day is essentially wasted getting acclimated to the time changes and getting over the trip.
Traveling Vienna--Venice is rather difficult in that it is farther than one would think. Only Austrian Airlines flies the route. Train passengers have to switch to a bus and back to a train on the shortest route to Venice. Taking the train West to Salzburg, thru Innsbruck and down to Venice often requires a transfer of trains in Verona. And that route is a very long train ride.
Vienna is a great European city in the continental style and worthy of the whole time you have on your trip prior to being in Venice. Taking in Hallstadt, Salzburg and Innsbruck just don't work well for the time available. I would skip them and take the shortest distance train going south Vienna-Venice.

Are you driving the whole time? Coming back to Austria to turn in the car, or dropping it in Innsbruck and taking the train to Venice?

It doesn't make any sense to drive anywhere. All these cities are well served by fast, clean trains, which will be faster than bumbling around a strange place in a rental car with a standard transmission. I personally don't like to sleep only one night in a place, but you seem to be willing. This seems like a rather rushed visit to Austria, which isn't that close to Venice. You have not described your transportation from Innsbruck to Venice.

When we visited Salzburg, we flew Austrian Air across the Atlantic, and used them the same day to fly to Innsbruck (beautiful air views). We used the train to spend a few nights each in Salzburg and then Munich, before having our Vienna nights, and flew home from there. I mention this because it might be much easier to FLY to Venice from Vienna than from Salzburg. But I haven't checked what's available at either city.

Traveling Vienna--Venice is rather difficult in that it is farther than one would think. Train passengers have to switch to a bus and back to a train on the shortest route to Venice.

That is not true anymore. There are two direct RailJet trains per day now from Vienna to Venice (7:40 hours ride). There is an additional connection from Vienna to Villach, where you board a bus directly to Venice (8:00 hours ride in total).

Given the short time frame, and considering the required travel times to all the destinations you mentioned, I would suggest to stay in Vienna only before going to Venice, i.e. skipping Salzburg, Hallstatt and Innsbruck.

There are 4 direct flights between Vienna and Venice per day (one in the morning, two in the afternoon, and one in the evening).
Price about $180.
https://www.kayak.com/flights/VIE-VCE/2018-06-13?sort=bestflight_a&fs=stops=0

My thanks for the responses!
First we are flying from Paris to Vienna, so everyone will be refreshed when we land.

I basically have 3 goals:
1. See Vienna
2. See some Castles
3. See the Austrian Alps

My theory about going from Vienna to Innsbruck is Innsbruck is much closer to Venice (It is about 4 hours).

Is there a way to see the best of the Austrian Alps and stay in Vienna the whole time - if so I would fly from Vienna to Venice.

Again, my thanks to all who have posted.