Where to stay

There are several decent hotels near the train station. What do you want - bare minimum? comfort? Do you have a price range?

I've stayed here, in Mainz, at Hotel Königshof, when I had a flight out of FRA in the morning. It has pretty good prices for the airport area and a GREAT breakfast. It's about a 30 min S-Bahn ride from the airport to Mainz, and Königshof is right across the plaza from the train station. You don't have to go back to the airport or into Frankfurt to catch a train to Paris; there are connections from Mainz to Paris with changes in Mannheim, Karlsruhe, Saarbrücken, or Köln.

I'd say if you plan on taking the 8:57 TGV direct to Paris, a hotel near Frankfurt Hbf would work best. Easy S-bahn to/from the airport. You can do a bit of light touring in Frankfurt, early bedtime for jet lag recovery and ready to go in the morning.

You should check the Frankfurt hotel rates before you decide to stay in Frankfurt. If a trade fair is in town, the prices skyrocket. This spring I stayed (2 times, 3 days each) in a 5-star hotel just off the main shopping street. The per night cost was about $112 per night (using Hotwire). I checked the prices for the same hotel on Expedia for Sep 17 & 18, $996 per night for the cheapest room. I don't know your situation, but can think of 3 options: (1) look in another town, (2) try airbnb, or (3) change the arrival date. BTW, why don't you fly to Paris direct instead of going to Frankfurt and then training to Paris?

I think you've already solved the hotel issue, if you're the same as this reader: https://community.ricksteves.com/travel-forum/germany/help-looking-for-a-place-to-stay-in-franfurt.