Train from Lucern tp Paris

It seems to take about 5 hours with a change in Basel. That is pretty easy.

https://www.bahn.com/en/view/index.shtml
https://www.sbb.ch/en/home.html

It is indeed quite easy. The Swiss Railways website or app (SBB) will even give you the number of the platform where you need to catch the train in Luzern and in Basel.

Easy. There is one connection every 2 hours, one change at Basel. Look up train times on the SBB website: https://www.sbb.ch/en
Typical timings:

Luzern dep 06:54, Paris Gare de Lyon arr 11:38
Luzern dep 08:54, Paris Gare de Lyon arr 13:38
Luzern dep 10:54, Paris Gare de Lyon arr 15:38
Luzern dep 12:54, Paris Gare de Lyon arr 17:34
Luzern dep 14:54, Paris Gare de Lyon arr 19:38
Luzern dep 16:54, Paris Gare de Lyon arr 21:38

I hope that you are spending time to visit Lucerne, as well. But if you don't have time there, then you could consider dropping the car in Interlaken (Hertz or Europcar locations). The train connections will be similar.

And in what country will you pick up your rental car? If you drop it in a country different from where you picked it up you will be facing some hefty cross border drop charges. Switzerland is not a country where you need a rental car.

Thank you for all the help! I have traveled in Europe many times on my own, but this will be the first time with my family, and we have quite the trip planned. We will be spending 15 days between Germany, Austria(day trip), Switzerland and France. Most of it will be driving(Frankfurt to Switzerland), but we will be taking the train from Lauterbrunen to Paris and Paris back to Frankfurt. I have a lot planned, but also I am ready to drop a few things if it gets too hectic and the fun factor goes missing.

Most of it will be driving(Frankfurt to Switzerland),

Virkus, as Laura said picking up a rental car in Germany and returning it in Switzerland will probably be expensive.. Somebody has to drive it back to Germany. Have you a cost estimate for the one-way surcharge?

but we will be taking the train from Lauterbrunen to Paris

In your original post you said "driving from Lauterbrunnen to Lucern . . . . train from there to Paris". You can do either, you just have to look up times and buy tickets for whichever.
Unless you want to spend nights in Luzern. The rail line from Lauterbrunnen to Luzern (Brünig Pass route) is scenic.

and Paris back to Frankfurt.

You could just fly home direct from Paris, and avoid doubling back.

To avoid a huge car rental drop-off fee, you could either:
- Drop the car in Germany (Munich), and use trains for your Austria daytrip (Salzburg I assume?) and Switzerland
- Keep the car for Switzerland (more on this below), but drop it in Germany.
--- Option 1 is Offenburg, a short train ride across from Strasbourg where you catch TGVs to Paris.
--- Option 2 is Weil am Rhein or Lörrach which are right next to Basel and have branches of Europcar, Hertz, Sixt and possibly others. But you'd have to make your way back to the Basel SBB train station, which can be awkward (I guess a taxi would be needed).

I strongly favor NOT having a car in Switzerland.

Or, car drop off points
If coming from Munich, trains are not good towards Switzerland. Salzburg or Innsbruck to Switzerland, via ÖBB (Austrian railways) is better.
But you could also drop off the car in Konstanz, which has good Swiss train connections.
Or, as Balso says, Weil am Rhein or Lörrach, both of which are suburbs of Basel. You could take a taxi into Basel, or a local train (frequent), Weil am Rhein has trams running to Basel (city centre and SBB station) every 15 minutes.
I have even walked from Weil am Rhein, along the Rhine, via France, to Basel city centre takes ~1 hour.

I think for us, it would be best to drop the car in Luzern even though it will cost us more $$. Thank you for all the different options though!

You'll also need to buy a Swiss Vignette for the car if you decide to stick with your plan. Cost is 40 CHF and you can buy it at the border or many gas stations near the German border. Be sure to attach it to your windshield per the instructions.

If your day trip to Austria includes driving on any Austrian motorways, you will need an Austrian vignette as well, but they have short term ones for not too much. If you stick to secondary roads, you don't need one.

Thank you for the reminder of the Austrian and Swiss driving passes! We are only doing a day trip to Salzburg from Garmisch, but will be in Switzerland fro 3 days.

The 10-day Austrian vignette is 9.20 EUR. It might be time consuming to get to Salzburg from Germany using secondary roads.