Paris to Frankfurt

I know of no direct (no change) train from Paris to Frankfurt airport, but there are a lot of one change TGV/ICE connections taking a little over 4 hours, usually with changes in Karlsruhe or Mannheim. Look them up on the German Rail Website. Use just "Paris" to get any train station in Paris and "Frankfurt(M) Flughafen Fernbahnhof".

"Is there a reasonable hotel ... at the Frankfurt airport?"

No. I guess it depends on your definition of "reasonable", but IMO, there are not. That's why I always stay here, in Mainz. Reasonable price, right across the Bahnhofplatz from the Mainz Hbf and the S-Bahn to the airport, great breakfast. I'm sure others will recommend places in Frankfurt itself, but I know of no place more convenient.

You can easily stay in Frankfurt and be at the airport in about the same amount of time as a shuttle would take. Some of the shuttles even charge you up to 5 euro per person, so you might as well pay for the train. Frankfurt is only 11 min. away and a train comes by every 15 min. The direct train from Paris will get you to Frankfurt in 3 hours 49 min.

As long as you are going to be in Frankfurt, consider sightseeing a bit in one of the most historic cities in Germany. Lots of museums to choose from, as well as several beautiful medieval churches, and other historic buildings, and some great farmers markets and the ever popular Klein Markt Halle. Hotels near the train station can be had for anywhere from 59-75 per night for a double. Check Hamburger Hof, Excelsior Hotel, Hotel Monopol, or Holiday Inn Express.

There's a Sheraton hotel attached to the airport, but it isn't cheap.

Most of the usual cluster of budget "airport hotels" are no more convenient than downtown Frankfurt, so you may as well stay there. Or in nearby Wiesbaden.

On the Bahn website (bahn.com) I see several trains each day from Paris (Nord or Est) to Frankfurt Flughafen with one change of trains in, for example, Cologne, Karlsruhe, or Mannheim. Use the booking section of the website to look at different journeys and times to see which fits your needs.

Re hotel: we stayed at the Hilton Garden Inn last year. It is near the airport train station and part of the airport complex at Terminal 1 so one simply walks to the airport. If you are leaving from Terminal 2 you can take a shuttle bus there from T1. This eliminates extra transport issues the day you are flying out. Suggest you check the hotel's website to see if they are "reasonable" by your definition. Have a great trip!