From Sarlat-Montpellier-Nice , how to get there w/out renting a car

I would switch things around a bit:

Paris
Sarlat (ideally with the Paris-Souillac train arriving at 13:30 followed by the bus, otherwise via Bordeaux)
Bordeaux
Fly to Nice
etc.

That way you solve the issue of going to Nice.

I also think you could shave a night off Paris at the end, and add it to Nice.
Do rent a car in Sarlat!

And I'm not sure Rocamadour is the must-see while in Sarlat. It's very touristy. I'm not super knowledgeable about the area so I won't make specific suggestions, but I know that there are tons of other villages to see. Beynac is one, there's Domme too.

If you don't rent a car, you only have two "easy" options to get from Sarlat to Nice:

1/ By train with two connections, first in Bordeaux and then in Marseille. The journey takes between 12.5 and 13 hours.

2/ By taking a train to Bordeaux (2.5 hours), then transportation from the train station to Bordeaux airport, and finally a flight from Bordeaux to Nice (1 hour 20 minutes).

You can also find a way to reach Toulouse (which isn't possible by direct train or bus). From Toulouse, it's another 7 to 8 hours by train to Nice with a connection in Marseille.

Keep in mind that in the Dordogne, 90% of the region's attractions are only accessible by car. Once in Dordogne without a car you are entirely dependent on the few public buses, which are not intended for tourism but mainly for school transport and some local workers.

You won't be able to get from Sarlat to Rocamadour without a car, and spending the night there is pointless, it's just a huge tourist trap for organized groups.

Balso's switch-up is a good idea. Train to Bordeaux (TGV just over 2 hours), spend 1 1/2 days (your 2 nights) there. Rent the car at the airport when leaving Bordeaux (a tram goes there from the city and it is easier to leave Bordeaux by car from the airport than from the train station in the city). Drive to Sarlat, visit the area for your 3 nights, drive back to Bordeaux and drop the car at Bordeaux airport and fly to Nice. The drive between Bordeaux and Sarlat is about 2 1/2 -3 hours each way. An early start gets you to a flight by mid-afternoon and into Nice in time for a nice dinner and stroll in the old town or along the beach.

Re your Dordogne segment. To me, Rocomadour is one of the least attractive options in the region and not super close to Sarlat. With 3 nights (2 full days, you could visit 2 caves in one day and the Castle at Beynac or Castelnaud, plus a market at one of the towns or at Sarlat (big ones there on Wednesdays and Saturdays. I see Oct 14 is a Wed.) Lascaux IV would be a good way to learn about the period and also enhance your visit to any other cave. Both of the castles are a real treat, historically and in their presentations. And they each have wonderful views of the river. You don't need a whole day for Font de Gaume and the tours are limited in number and size. Go in the morning, have lunch somewhere, then visit a castle. See Lascaux IV a different day and go to a market. There will be time probably to drive to another lovely town. Domme, perhaps. Or gardens such as those at Marqueyssac. Play with your schedules. Also, a drive to and through some of "the most beautiful towns in France" is easy here. There are many!

You will have a wonderful time! Enjoy!

Thank you all for your sage advice. Will check into flying to Nice.