"The plan for the second day is to see the Burg Eltz castle and Cochem."
Zack - there's no problem visiting Burg Eltz on a day trip from Mainz and making a stop in Cochem. That's a lot more doable than "exploring" the Mosel Valley by car, which I apparently took more literally than you meant it - and which requires far more miles and hours than a quick glance at the map might suggest.
Viamichelin shows the drive to Eltz a little longer than your source does, but either way the car should be a time-saving asset. The walk from Moselkern is not difficult - I was pushing 50 when we went there with my 75-year-old in-laws, and they handled it fine and enjoyed it. It's a nice approach as long as it's not hot and humid, or rainy. But we had an apartment in Cochem; that walk from Moselkern + the additional train travel time required from a Mainz base will chew away at your day.
"Our plan is one day to take a cruise from Bingen to St. Goar and then check out Bacharach and Oberwesel on the way back."
For a scenic walk, this day looks more promising to me! You might want to do all or part of the Rhine Castle Trail between St. Goar and Oberwesel - very dramatic views from the trail there. This shot of the Loreley cliffs was taken just a bit south of St. Goar from the "Maria Ruh" lookout. This one takes in the town of Oberwesel (background) and the river gorge from the Günderodehaus, a little detour off the same trail where you can get refreshments.
In Oberwesel, the town wall walk involves less elevation gain and is also very rewarding.
Lots of people have done the walk from Moselkern that's outlined in Rick's book. For some reason - probably because Rick hasn't investigated them or emphasized them in his travel materials - I have not "bumped into" people on this forum who have experience with these walks on the Rhine.