We need suggestions on the best ways to travel from Salzburg to Lucerne. Is driving a rental car an option (I see a suggested driving routeof 376 miles, which is a bit much, and I'm not finding rental companies with a pickup in Salzburg and drop-off in our final destination of Zurich)? What about trains (Eurail is a 6+ hour trip with only 15 minutes between connections -- is that enough time to make the connection?)? Air? Some combination?

Driving is probably not an option if you can find a rental agency as you will have to pay a stiff cross-border drop-off fee. Unless you want to do some touring, the train is a better option.

If you mean that your 15 minute connection is in Zurich, you need not worry about it. There are trains between Zurich and Luzern almost every half hour.

The car would incur a big drop charge--not a good idea.
Check the bahn.de site for rail connections---your likely best option.

Your change time in Munich is 17 minutes and is OK. If the train is late just catch the next one to Zurich. You don't need reservations on these trains. The scenery will be nice. See the DB site for a route map. http://tinyurl.com/c9jp54 The ticket is only €59.

Oops, I missed the city, by a few miles!

Driving makes sense if you're going to use the car for other days/day trips or whatever. Just to get from one place to another I would hop on a train.