The schedule is indeed a bit tight, but no reason for not visiting Austria ;-).
Given your time constraints, I would skip Hallstatt and focus on Vienna and Salzburg. If you focus then on Vienna and Salzburg it is not necessary to rent a car.
I would divide the time equally between Vienna and Salzburg, or even spend a bit more time in Vienna. There are plenty of things to do and to see.
If you want to get the most of this 7-8 days, arrange everything beforehand, i.e. booking trains and hotels. This will allow you to fully concentrate on the "essentials".
Also note that "Eagle's Nest" is located near Berchtesgaden (Germany), some 30-40 kilometers from Salzburg. It can be reached by public transport. This visit will take at least half a if not a day.