Not sure if you have 6 nights or 5 nights and six days.
You can do this with some careful planning.
WHEN are you going? That makes a huge difference. Many of the less urban areas are virtually deadsville in the off-season, sights and tourist services inaccessible.
I'd suggest 2-3 nights on the Rhine (by train, St. Goar is the best spot), 2-3 nights on the Mosel (Cochem) and 1 night in Rothenburg (too far for a daytrip.) It would be possible to daytrip on both Rhine and Mosel for 4-5 days from Koblenz, but Koblenz isn't as scenic or interesting a base as the other towns. Still, in the off-season, Koblenz might be better since it's a real city and will have some services available to you.
You may be able to end your trip on the Rhine; from St. Goar or Koblenz it's not far to FRA. Just depends on when your flight is. If that doesn't work, stay in Mainz on your last night - just 20 minutes to FRA from there.