trains between Innsbruck and Paris

Rail Europe is just a travel agent. They are not the ones that run the trains.
You need to familiarize yourself with train travel. Start reading at www.seat61.com

And a good start is to look up connections at www.bahn.de The route via Zürich can probably be bought from www.sbb.ch as a single ticket.

There is one stop in Zurich but there is only 14 minutes between trains. Would this be enough time? The next train from Zurich would be five hours later.

This is not correct. There is a train from Zurich to Paris every two hours. The trains from Innsbruck to Paris via Zurich are deliberately timed for a short, but manageable layover.

Compare the prices for that route on both the Austrian and Swiss train operators' websites.
https://www.oebb.at/en/
https://www.sbb.ch/en

Another possibility is backtracking from Innsbruck to Salzburg in the evening, and then catching the NightJet (sleeper) train to Paris, departing shortly after 9pm.

You can actually book this on www.bahn.de. Their international sales site is quite good and can even book tickets for trips that do not touch Germany at all. The SBB site is sometimes problematic with US credit cards.

The short connection is fine. This is the railways trying not to waste your time. If a delay makes you miss your connection you take the next train. So maybe do not book the last connection of the day...

Please, please, please do not use Rail Europe as they do not operate any trains - they are simply a travel agency.

Look at this link from the Man in Seat 61 which has very specific instructions for this route - https://www.seat61.com/international-trains/trains-from-Paris.htm#Paris-Vienna

14 minutes is more than enough time to change tracks.

Also, why not fly?

Also, why not fly?

Because there are no flights from Innsbruck to Paris?

But there are airports in Paris and an airport in Innsbruck? One stop, 4 hours total travel time, minus getting to/from airport. Train takes about 8 hours, minus getting to/from train station.

One stop, 4 hours total travel time, minus getting to/from airport.
Train takes about 8 hours, minus getting to/from train station.

So downtown to downtown the train is as fast as the plane. But the train will be more pleasant.

Use bahn.de to check times and book your tickets. 14 minutes should be enough time to change in Zurich Hbf from your Railjet (Austrian express train) to TGV (French express train) and if there is a delay you should get the guard on the Austrian train to stamp your ticket for you to indicate that there were problems.

The route from Innsbruck to Zurich is very scenic, and there are some nice bits between Zurich and Paris, especially as you pass through the Franche-Comte lakelands. (ETA: Sorry, I was thinking of the line from Geneva to Paris, Zurich to Paris isn't quite as scenic, but it's still more pleasant than flying.)

Use bahn.de to check times and book your tickets.

The route from Innsbruck to Paris does not touch Germany. So it might not be favorable to buy the tickets from the German train operator.

When buying tickets for a border-crossing train it is always advisable to check the prices offered by either train operator. In this particular case you have the choice between Austrian, Swiss and French railways.

The route from Innsbruck to Paris does not touch Germany. So it might
not be favorable to buy the tickets from the German train operator.

DB has a separate site for international bookings: https://www.international-bahn.de/en/
And it is actually a pretty good site. For example, it managed to book Basel - Luxembourg for me, a trip that involves only SNCF trains, but that SNCF itself could not book for me.

It can book Innsbruck - Paris, and will automatically book the part Innsbruck - Zurich on OBB, and Zurich - Paris on SNCF.