Considering car rental

You really don't need a car in Salzburg or Vienna, and parking in the historic area would be quite difficult. If you want to visit some of the smaller towns or villages and Melk Abbey after leaving Salzburg then a car makes more sense.

I am doing a similar itinerary this summer. I can't say what works best - I haven't done it yet! - but I looked at similar options to you. What I chose was to rent a car upon leaving Salzburg, drive it to Halstatt, stopping at some lake district places TBD on the way, keep it in Halstatt (I might not drive it during those days, or I might take it to the mountain nearby), then drive it to Vienna, returning it on arrival in Vienna. Between Halstatt and Vienna, I may stop at Melk or elsewhere on the Danube, or maybe stop closer to Hallstatt for hiking. In that case, I'd do Melk as a day trip from Vienna by train.

Your option 1 is reasonable, but for me it seemed that there's enough to do in Salzburg/nearby by bus so that an extra day trip wouldn't be necessary.
Your option 2 is as I noted in point 1 - I don't feel I need car during my Salzburg time.
Your option 3 is what I chose, basically.
Option 4 works - technically there is no need for a car. If all you want is Salz. - Hall. - Vienna, it looks easy to do by train! And Melk/Danube is an easy day trip from Vienna by train. However, I wanted to go into natural areas, go hiking, that kind of thing. I like the flexibility - timing hiking/walking to the more limited bus schedules to some areas can be tough.

Of course, I'm talking with no actual experience, just as someone who has researched this for my own upcoming trip!

You're facing exactly the questions I have asked myself. I will need to learn more about what is on the way from Hallstat/Salzburg to Vienna. Thanks.

I think it would a great idea to drive from Salzburg to Vienna and see Mauthausen and Melk along the way. It is not easy to get to Mauthausen except by car. You would not want to drive in Melk but it is small so you can park outside the town or abby and then walk through. Rick's guidebook talks about both these sights. The Wachau Valley is also a nice scenic drive.

We did a day trip out of Vienna to see the Mauthausen Memorial. I rented a car from Hertz at Vienna's Haupbahnhof or main train station. I brought my GPS from home because it had European maps and I am used to it. I chose Hertz because my credit card offered a discount and it was cheap. I had an International Driving Permit which made the rental process a little easier, I think.

I rented a Ford Focus wagon for € 59.39 when the Hertz office opened, drove to the hotel, and picked up the rest of our party.

Driving to Mauthausen was easy. The autobahn was safe and quick.

Mauthausen was truly incredible. The emotional impact of seeing where the suffering happened was unforgettable. I loaded the Mauthausen visitor app on my phone which provided a lot of information. After visiting the camp, we drove down the hill and turned up a small road to the bottom of the quarry. It was a short walk from the small parking area to the foot of the stairway of death.

After returning to Vienna, I dropped off my passengers at the hotel and drove back to the Haupbahnhof. I put € 26.80 worth of diesel in the tank before parking it. I took a lot of pictures just to be sure and dropped the keys in the box.

So for less than the price of one guided bus ticket, the three of us made the trip. The roads from the Haupbahnhof to the outside of town are easy to drive. Our hotel in the inner part of Vienna was not. I would recommend that everybody goes to pick up the car for that reason.

I hope somebody finds this information useful.