I'm not clear exactly what your itinerary is at this point but gathering up the suggestions I think it's something like this...
- Munich - 2 nights, Salzburg - 2 nights... OR Munich 4 nights w/ day trips to Regensburg, Salzburg
- Somewhere between Salzburg and Mainz - 2 nights (maybe Würzburg w/ day trip to Rothenburg)
- Mainz - 2 nights (w/ day trip along the Rhine)
Except for the Rhine, you are spending most of your limited time in cities, all of which are served nicely by train. But the Rhine is also well served by the rail system - trains run hourly or better along both river banks. Your cousin says, "...renting a car is easy and allows you to see more of the countryside." That could make sense for a more leisurely trip. But your travel pace is swift, and it's a very big stretch to imagine yourselves roaming at will over hill and dale when you're covering so much territory in 8 days. In fact, you will spend extra time looking for (and paying for) parking, picking up and dropping off the car, gassing up, making potty stops, driving in sometimes-heavy city traffic, obtaining environmental stickers and vignettes...
So on the way to Salzburg, or to Rothenburg, for example, will you see more countryside from your car window, or from the train window?
Train prices for 2 adults:
Munich-Regensburg-Munich: Servus ticket (day pass) €26
Munich-Salzburg-Munich: Bayern ticket (day pass) €31
Munich-Würzburg: pre-purchased saver fare for long-distance trains €48
Würzburg-Mainz (only 1:49 travel time by direct IC train): pre-purchased saver fare €38 (I checked June 5-8 for the 10:29 journey.) Regional trains take 2:35, cost more (€52.)
Mainz-Middle Rhine Valley-Mainz: Rhineland-Palatinate ticket (day pass) €29 (also covers certain ferry crossings.)
Mainz-FRA airport: €10
Plus you have a local MUC to Munich fare. That's roughly €200 total for two. I think you will find that gas alone for the same journeys will price out at around €130. Then you have rental fees, insurance, parking fees, stickers and vignettes.
There are cases when a car makes very good sense, but I don't see yours as one of them.