"I'm thinking we could go from Brugge, stop to see the cathedral in Cologne, then stay the night in Mainz... but I had trouble..."
It works for me when I use Mainz Hbf as a stopover. Of course, you don't say what the travel date is. You said you want to travel via Ulm, right? I tried this data:
Departure date: Thursday July 9.
Start: Brugge (9:08)
Destinaton: München Hbf
Then input your stopovers...
Via #1: Cologne (2 hrs. stopover time)
Via #2: Mainz Hbf (14 hours stopover time)
Result:
1.) Trip from Brugge (9:08) with a change in Bruxelles-Midi, arrive Cologne 13:15 for stopover
2.) Leave Cologne 15:53, ; arrive in Mainz at 17:38 - travel via the scenic Rhine Valley via Bonn and Koblenz
3.) Leave Mainz on July 10 (7:46) with a change in Mannheim; arrive Munich 11:27 (ICE 591, stops in Stuttgart, Ulm and Augsburg on the way.)
Saver fare is available for this route.
Don't worry about the reservation. It's not required for this trip. You can buy this ticket without one and buy it later if you really want to reserve a seat.