"How should we plan our time to hopefully see Bavaria, Munich and Salzburg?...Recommended home base in Bavaria?"
First of all, let's clarify the geography. Bavaria is a huge state that stretches almost to Frankfurt in the northwest and includes Munich and Rothenburg. If one wished to see all of Bavaria, a single base won't work - it's just too large.
"We can rent a car if necessary." Don't worry - it's not, as you can see from this rail line map. If by chance you pick a destination without a station, it's likely tied to the rail system with a bus connection.
Recommended home bases: It's possible to cover much of northern Bavaria from a base in Nuremberg (a major train hub and quite a nice city.) In the south, Munich (or maybe Augsburg) works pretty well. But neither of these will cover all destinations. Not that that should be a concern - you don't have time to see EVERYTHING anyway. But if your point is that you'd like to minimize the number of places you stay, I'd begin looking at destinations within traveling distance of Nuremberg and Munich and consider 3-4 days in each. Some suggestions:
NUREMBERG BASE day trip possibilities (none of these destinations is more than 1.25 hours away.)
Bamberg - Würzburg - Iphofen - Bayreuth - Rothenburg - Bad Windsheim (great open-land museum) - Regensburg
MUNICH BASE day trip possibilities (up to two hours away)
Augsburg - Landsberg - Landshut - Prien and Herrenchiemsee - Füssen - Mittenwald - Garmisch-Partenkirchen - Salzburg - Dießen - Herrsching and Andechs Monastery - Regensburg - Nördlingen - Burghausen
Getting to any of these places is easy when you buy a regional day pass. There are dozens of different regional day passes good for regional train travel within Bavaria, but one day pass that covers all of Bavaria - the Bayern (Bavaria) Ticket - which includes Salzburg and costs €28/day/couple. The other day passes are valid within various subregions of Bavaria and cost less. For example, the day pass for the Nuremberg region (called the Tagesticket Plus) costs €18.70/day/couple and gets you to all those destinations listed above but Regensburg (use the Bayern ticket instead.) Some day passes like the Bayern Ticket have hours restrictions (like after 9 am if it's a weekday) but others do not (the Tagesticket Plus, for example - which has an additional perk - it's valid for both Saturday AND Sunday if you buy it on Saturday.)
So I'd start by looking to see what interests you in some of the destinations named above.
(Don't mean to diss your plans but do understand that Rothenburg is one of Germany's most touristy and tourist-packed enclaves; its streets are routinely overloaded by the 2.5 million tourists it gets annually (tourism is the only game in town,) everything's in English, and as Rick says, "Rothenburg is well on its way to becoming a medieval theme park." If my best friend had 7 days to see the real Bavaria, I'd send him elsewhere.)