Road trip starting in Germany

If you haven't already done so, go to websites such as Michelin.com (for the road part of the trip) and train websites (for the train part) that will help you to get a realistic idea of the travel times.
Some people find that Europe is bigger than they thought and that travel times were longer than they thought.
And then have a great trip!

I have done that and love that website. It's mostly why we have decided to do more trains than driving.

Yes, unlike western N. American, where there are few trains, in Europe many locals use the trains, and there are good reasons for that.
But there are a few areas that cannot be adequately explored by public transportation, and the trick is knowing whether any of the areas on your itinerary are one of those.

Right, exactly. We have driven in Europe many times, just recently from Netherlands to Belgium and you realize the roads are very very busy with so much traffic, so much construction that the driving times are just rough estimates. So training is less stressful there for sure especially with so many stops.

Right. As you probably know, mountainous areas and other remote areas that are not adequately served by public transportation--but which may be on your itinerary--are the areas where a rental car makes the most sense.

Such as, kent?

As you already know, the major cities are generally well served by trains.
Mountainous areas (you mention them in your itinerary) are areas for which you would want to do extra research, if possible, to try to determine whether the areas you want to visit in these more remote areas are adequately served by public transportation. Local buses can be cost effective, but sometimes require more time, slowing you down more, and sometimes require more knowledge of the local bus system than is feasible for a traveler to acquire ahead of time.

Right. We were thinking of renting a car to go to Lichtenstein for sure.

I suggest you use the German Rail website to look for train schedules.

I see only one night train connection from Luzern to Salzburg, and that has an hour by regular train to Zürich, and then the night train from Zürich to Salzburg in less than 6 hours, getting to Salzburg at a little after 4 in the morning. I don't know that the night train would be worth it. You can go during the day by RailJet from Zürich just as fast and see the scenery.

Not sure it would be worth taking a car to Liechtenstein (although I don't know how difficult it is to get around in Liechtenstein). The rail connection to Buchs, across the river from Liechtenstein, only takes a couple of hours from Luzern.

I know, it's true. The overnight train does not sound good really and getting in at 4 AM is ridiculous. And there is beauty to be seen so railjet it is! Thank you so much.

Is a 15 minute transfer and then a 4 minute transfer between trains in Switzerland something we should even consider risking?

Those short transfers are no problem if they're toward the end our your connections and there's a later train that can also work. If you're connecting to a long-distance, reserved/nonrefundable, or overnight/once-per-night train, then give yourself more time. The DB site does allow you to request longer connections at either the start of the search or even after you get schedules (click on the connection time).