Paris-Loire-Dordogne

Why are you going to Avignon? To get to Avignon from most places in the Dordogne area will take at least 5 hours by car and about that much or more by train. If you are going to Avignon just to get a fast train back to Paris, you are wasting your time and that last leg is what would make this trip too much to get done in nine days IMO. If I get your description correctly, you plan on two days in Paris and three days in each of the Loire Valley and the Dordogne. Driving is very doable in the Loire Valley and without a car exploring the Dordogne will be really limited. If you want to start with the Paris, I'd take the train to Saint Pierre des Corps and rent a car there. I'd drive from the Loire Valley to the Dordogne (about 4 to 5 hours) and after spending the remaining days in the Dordogne drop the car off in some place like Libourne or Brive-la-Gaillarde and take a train back to Paris. As it is, don't forget that travel time is going to eat up a big chunk of 2 of your days i.e. the day that you travel from the Loire Valley to the Dordogne and the day that you travel from the Dordogne to Paris. For me personally, I like to go to the furthest spot first and work my way back to my departure city so I would go to the Dordogne first and then make my way to the Loire Valley and end with two days in Paris and so avoid the last day rush to Paris to catch my flight.

If you haven't already booked your flights, consider flying into Paris and out of Bordeaux or Toulouse (or vice-versa). This saves backtracking to Paris, and may not even cost much more than a simple round trip.

To find these flights, use the "multi city" option on Kayak, Air France, etc. Don't try to book two one-ways, as that's much more expensive.

Yes, I would pick up a car in the Loire and keep it for the Dordogne and Provence regions, to help you reach more smaller towns and sites. There should be no particular car rental "penalty" to do it that way.

Then you'd just need two train tickets, which you can purchase ahead for better rates (cheapest rate called "Prems" is on sale from 3-4 months before your travel dates and the SNCF takes PayPal payments for these).

I skipped the Loire and took the TGV from Paris to Bordeaux. Rented a car and drove through St. Emilion to the Dordogne (set the GPS to avoid toll roads and took a scenic route). After three days in Dordogne we drove to Provence stopping for lunch in Carcassonne. When I go back I'll probably stay in St. Remy and then drop the car at the train station in Avignon before the TGV back to Paris. I'd look into flights to or from any of your starting or ending points but if you are committed to flying into and out of Paris then my suggestions will be logistically easy.