I agree with Sam, it would be useful to know how many nights you are staying in each location.
Train from Munich to Zurich (3)
Train from Zurich to Lucerne (4)
Is that overnight in Zürich? (how many nights?), or Munich to Luzern in one day - in which it counts as one rail trip, not two. I would recommend not staying overnight in Zürich, but continuing to Luzern.
Train back from Lucerne to Zurich (5)
Train from Zurich to Paris (6)
Is that overnight in Zürich? (again?), or Luzern to Paris in one day - in which it counts as one rail trip, AND there is no need to go via Zürich, there is a shorter route via Basel.
When looking up train routes and times, use the DB website ( https://www.bahn.com/en/view/index.shtml ). There is no need to second guess the route and trains needed. If you want to get from A to B, just lookup A to B, and the website will find the quickest route and tell you how many trains are needed.
As to a Eurail pass - almost certainly not.
- Munich to Salzburg day trip is cheap with a Bayern Länder ticket, which gives extra savings for groups.
- Munich to Luzern, you get discounts for advance purchase. I looked at a random date in May and saw tickets from €39.90, a lot cheaper than a pass.
- Luzern to Paris, also discounts for advance purchase.
- Paris to Munich, I saw tickets for €39, a lot cheaper than a pass.
What train or boat trips will you be doing from Luzern? It may be worth getting a local pass to cover these day trips, called a "Tell Pass". Details here: http://www.tellpass.ch/en/summer/