It doesn't seem reasonable to visit southern Germany sights via
train, esp. if one wants to explore the small towns.
Germany is different. Visiting small towns by train might seem counterintuitive if you've found small towns hard to reach by train in other European countries, but Germany's rail system makes it quite easy. Bavaria's rail network is incredibly dense.
I use your base-town-with-day-outings strategy all the time - it's a great way to see a country like Germany. The hard part is picking base towns which are appropriate for your day-outing destinations. As geovagriffith and sla019 have indicated, those destinations are spread too far apart for a single base town to be effective without a lot of extra ground travel. You are visiting only PART of southern Germany - Bavaria - but Bavaria is much too large for just one base town to work.
With your interest in smaller towns in mind, I took a look at some prior posts to learn about your interests...
We're not that much into museums - more into walking through charming
towns with narrow streets, shops, restaurants, and such, and
interesting natural sights.
Your base towns do not have to be cities like Ingolstadt (140,000) or Munich or Nuremberg. Stay in small towns if you prefer those - well-located small towns of course.
SOUTHERN BASE: sla019's Murnau am Staffelsee could be a very good choice for you. From there, day trip to the Zugspitze, to Mittenwald (small town) and to Oberammergau (small town) as well as into MUNICH (which you were planning as a daytrip destination from Ingolstadt, I believe.) Alternatively, MUNICH might be visited while you are in transit between Murnau and your base town to the north...
NORTHERN BASE: This is such a wonderful area for several days... For day trips to Nuremberg, Rothenburg, and small towns nearby, I'd suggest one of these small-town travel bases...
Bad Windsheim (Franconian open-air museum)
Neustadt-an-der-Aisch (handsome town, zero tourists)
Ipfhofen (wine/art town)
Würzburg and the smaller Main River wine towns of Marktbreit and Ochsenfurt would also be possible to visit from one of these base towns.
Every one of the towns mentioned has regular, frequent and dependable train service. I've used these routes many times - they are scenic, comfortable, relaxing as a rule, and they make for very inexpensive getting around. I recommend the trains without hesitation for your entire itinerary within Germany. If you want a car for France, pick it up in Germany and drop it when you re-enter Germany at the station of your choice. Most rental companies have a drop-off fee of €0.00.