In my humble opinion, Halstatt isn't worth 3 days at all - its not even worth 3 hours. I might have an unpopular opinion, but I think it was maybe the most overrated place I've ever been in Europe. Like everyone else, I wanted to see it because its so popular with its "most beautiful" rankings, but I honestly wish I had skipped it, and let the imagined magic of Halstatt live on in my head.
The village is absolutely jam packed with tour groups, and instagrammers - I don't normally mind this, but the locals were some of the rudest I've met in Europe (they seemed annoyed at having to deal with tourists, who are the only source of income for them). But most importantly, you've probably seen so many pictures of it, that when you arrive, you really feel like youve already been there. I've never had this feeling anywhere else. Once you see the view from the lake, the small town square, and the waterfall, youve see seen it all. There's really not much else to see - its a small village that takes about 1 hour to walk around. The churches arent particularly interesting at all compared to what else is available in the region. I trule dont think the village on the whole is worth more than about 2 hours max. Additionally, I dont think its particularly conveniently located as a base to go anywhere else, so I wouldnt even dedicate a night there. That being said, I drove there - you can rent an electric boat and go out on the lake yourself to get the nice views of the village.
Rather than Halstatt - I think you'd be better off in my opinion going to see Neuschwanstein and the absolutely incredible Linderhof castle. In my opinion, this region is a much better way to enjoy precious vacation time. Bavaria's scenery is far better than that of the Austrian Salzkammergut region too (I know a post above disagrees with me on that). Rent a car from Munich, and add the Pilgrimage Church of Wies (en route to Neuschwanstein) and Ettal Abbey (which you can see after Linderhof. You might be able to have a quick look at the small towns like Fussen or Oberammergau, the latter of which can be seen by a quick stroll. You can do this as a day trip from Munich - distances aren't very far in Europe - or pace yourself and stay near Oberammergau for your extra night. The amount of "must see" sights you can see in this small corner of Bavaria blows Halstatt out of the water.
3 nights in Munich is good, so that you'd have at least 2 full days there- more if you like museums. Salzburg city can be seen in about 1 full day (It's a beautiful city, but quite small; do not miss the Golden Room of Hohensalzburg Castle) - and then I'd add about 1 day for nearby Berchtesgarden (that 1 day is assuming you have a car, and don't want to do any hiking), where you can see: Parish Church of St. Sebastian, WW2 sites like the old Berghof ruins, The Eagles Nest (you really dont need to spend much time at the time at the top), and +/- Wimbachklamm. With more time, you could see Koenigsee.
As for the Romantic Road - the part near Neuschwansteinand the part near Berchtesgarden are the most interesting.
Regarding car rental vs train - I generally prefer cars because you have so much freedom to do what you want on your own schedule, and you can easily visit places that would otherwise require lots of coordination on public transportation. For example, going to Wies to Neuschwanstein to Linderhof in a day would be time-consuming on public transit (I dont even know if Wies is served by transit). With a car you can easily do it yourself because you dont need to coordinate schedules or wait at stop/stations, and you eliminate the anxiety of having to always keep on eye on the time to catch the next bus or train. The only time I use trains in Europe is for high-speed intra-city services, and public transport within a metropolitan area. If you're going to rural areas, my general rule is to always get a car (OK Mobility is a Spanish chain that has good rates in Munich).