Aside from Trier, which I haven’t visited, I have actually driven to each of these places over a combination of trips. (Okay, I didn’t actually drive to Murren, but I did visit Lauterbrunnen for a week with a car.) Connecting these dots is very doable, especially with the amount of time you have. As others have said, it is possible to do this whole trip by train, with the odd bus ride in some cases, but I will address the car option.
Picking up a car at Frankfurt airport is very easy, but I wouldn’t recommend driving very far immediately after your very long flight. Do you plan on visiting Frankfurt for a few days? If so, pick up a car on your way out of town. If not, consider driving FRA to Bacharach. It’s just over an hour drive from FRA- about half the time as Trier- and is a good base to explore a few towns along the Rhine (easily done by boat, train, car or bike). You could even day trip to Trier from there, or make Trier your next stop. I’d allow four days for Bacharach and Trier combined. Baden-Baden and Freiburg are close enough together that I’d stay in one - preferably Baden- and visit the other on a day trip. Allow three days total, or four if you want to slip in a trip to Strasbourg. From Baden to Lauterbrunnen is at least a four hour drive, but not a difficult one (be sure to buy your Swiss vignette at the border, for CHF 40). Park the car at the multi-level parking lot at the Lauterbrunnen train station, and take the lift/train combination up to Murren (or stay in Lauterbrunnen). Allow at least 4 days here if you’re in to hiking, fewer if you’re not. From Lauterbrunnen to Fussen is another long drive, but you might want to give either Appenzell or Bregenz a short stop of a night or two along the way. I assume your reason for Füssen is to visit Nueschwanstein, so a night there is probably enough. I’d recommend going from Fussen to Salzburg (another vignette required for Austria) and staying 3 or 4 nights. Hallstatt is very picturesque, but you will by now have driven through a lot of equally picturesque places. If you must, visit Hallstatt as a day trip from Salzburg, but I’d say it’s expendable given what you already have on your itinerary. From Salzburg, drive to Munich and ditch the car (from Frankfurt to Munich, it’s an all-Germany rental). Stay in Munich for 4 or 5 days. Depending on how late into September your trip is starting, you’ll probably be in Munich during Oktoberfest. That’s a good or bad thing, depending on your tolerance for beer, crowds and higher hotel prices. Regardless, when you’ve had enough of Munich, take a train to Vienna to finish off your trip. Allow lots of time in Vienna, as there is an abundance of things to do there, and lots of worthwhile side trips you can take.
You have a great itinerary, and enough time to do it comfortably. If I were trying to do this trip in two weeks, I’d certainly do it as a RS tour, or use train and busses. But with 28 days, you can afford to linger longer in the places that interest you, and having a car will allow you the flexibility to get out and explore more. Enjoy!