where to stay between Freiburg and Frankfurt

"Baden-Baden has been suggested...is it a typical German town in addition to being a spa resort?" Depends on what you mean by "typical". Meaning, does it have a lot of post-war housing, a cluster of Mittelsand small industries and a Reve, Mediamarkt, Real, Aldi, and Lidl sitting on the edge of town? In that sense, yes, Baden-Baden is a typical German town. Of course, it's distinguished by its Thermes, the real casino (as opposed to the 24 hour shady little Spielotheks that nearly every German town seems to have), large concentration of restaurants and upmarket stores selling bling to Russian tourists with more money than taste. But comparing it to several other German spa towns, like Bad Homburg, Bad Hersfeld, Bad Kissingen, Wiesbaden, Königstein im Taunus, etc., I find it rather uninteresting. Freiburg to Frankfurt is about a 3-4 hour drive (traffic depending), so any place in between or a little beyond would be reasonable. Perhaps stay somewhere on the Mittelrhein and use your full day to drive around and explore? Heidelberg might also be a good choice.

"We are planning to rent a car for travel between Freiburg and Frankfurt." Beilstein is very tiny. It's hard to believe that both Baden-Baden and Beilstein are on your short list - they're almost polar opposites. None of the towns besides Baden-Baden lies between Freiburg and Frankfurt, BTW. Why must you go to Frankfurt to catch a train to A'dam?? Do you already have an advance-sale ticket that specifies travel beginning in Frankfurt, maybe? Bacharach makes sense as a mid-way stop between Freiburg and A'dam. If you HAVE to go into Frankfurt, I don't think I'd want to drive in from there - you're only about an hour from Frankfurt by train. With 2 nights, you'll find plenty to do in the area (castles, other villages, river cruise.) And northbound trains from Bacharach will get you to A'dam easily enough. If Bacharach ends up being your choice, I wouldn't see much point in renting a car; Freiburg-Bacharach-A'dam is a major train route. Freiburg-Frankfurt is too.

Of the towns you listed, my preference would be Bacharach. A small, quaint town with lots of lodging possibilities. You can take a cruise on the Rhine during the day and go to a local vintner at night for wine tasting. Read about the town in Rick Steves' Germany tour book.