Five Countries - 21 days

Your questions would take quite a long time to answer. You might do better to go to the bookstore or library and get some travel books where you can research your trip. Rick Steve's books are great.
If you have not purchased your flight tickets, you would do better to purchase a multistop ticket into Prague and home from Paris. You are probably underestimating the distances on the ground and backtracking is inefficient and expensive.
When traveling in Czech Republic and especially Austria, I see no need to visit an expensive Switzerland. I get my mountain fix in Austria in The Alps there.

If you rent a car in one country and drop in another, you will likely have a huge drop fee.
Also, having a car in some of those cities will be a liability, with parking issues. Especially Venice.

Paris has a wonderful METRO system, so you don't need a car there. Also, it makes it easy to get to the train station that you would take to go east. Gare de Lyon is the one SE of the city for trains to Lyon, South of France and Switzerland. Gare de East is the one NW of city and for travel to Germany.

21 days, sit down and pick your destinations and determine what you want to see in each. You many need to eliminate some, since just Paris is good for 4-5 nights. The RER is a separate rail system that goes from CDG airport to the city, also to Versilles, easy to get there.

Switzerland is beautiful, but very expensive, even compared to Austria and Germany. Vienna and Salzburg are wonderful, suggest minimum of three nights in Vienna and 2-3 in Salzburg (do you want to go to Berchtesgaden, Germany nearby?).

You should not need an international drivers license in France, Germany, Austria or Germany.

1. Don't fly in and out of Paris. You waste precious time and money by doubling back needlessly. Fly open jaws: in to one city, home from another.
2. Start by being honest with yourself. How many FULL, usable days do you really have in Europe, not counting the days you arrive and depart?
3. Be careful about trying to cram in too many places in too short a time. Especially traveling with a 79 year old. Remember that you will eat up significant time just with the basic daily needs, then when you add the time needed to pick up and move somewhere, it goes quickly. Can't tell you exactly yet but with your description of "five countries - 21 days" I'd wager that you are very probably trying to do too way much.
4. You should start with getting a travel guide book - Rick's basic Europe Through The Back Door is a good start.
5. Yes, you can drive all over Europe, but check specifics carefully, as it's common to be hit with a very large fee for renting in one country and dropping off in another. Often the best way to beat this is to travel mostly by trains, and rent a car where you really need one, so you avoid the injurious drop-off fees.
6. Get an International Driving Permit - at your local AAA club office.

That's a start... :)

Hopefully you have not purchased r/t tickets to Paris. You need to do a 'one way' trip, i.e. Paris to Prague or the reverse.

See no practical way to work a side trip to Venice into your trip.... its too far off the track of all your other places.

We loved Hotel Jungfrau in Murren. Splurge for a balcony room as the view is more than worth it.
Hotel Relais Bosquet is my favorite in Paris. Quite close to several items on your list.

As you refine and edit your plans, consider a "3 night rule". Spend at least 3 nights in places that interest you. You want to make sure that you are not spending all your time "enroute" vs. enjoying the places you are visiting. Also, even an active 79 year old will need some rest times.

You're trying to cover an awful lot of ground. Looks like you're trying to cover all your bucket list places in one trip. It would be a good idea to focus on priorities and your time or you'll be spending most of your trip in a car or train.
Almost everywhere in Europe can be reached by rail. So its all "doable". But whether any destination is a good use of your limited time and money is up to you. Venice is great, for example, but if it takes a whole day to get there, and a whole day to get to the next stop, maybe its not worth it to you.

Just addressing one of your questions:

Recommendations on specific trains to take out of Paris heading to Switzerland?

You don't have to identify a specific train. Its not like picking an airline. You pick your starting point and destination, and the website will provide you with the most efficient connections with times and costs. You pick the one that best fits your schedule. Use the country's website (SNCF for France, for example), or one like trainline.eu that covers multiple countries.

As others have mentioned, look for flights into one city and home from another. This is called "multi-city" and usually costs about the same as a round trip.

Definitely try to get the open-jaw flight - into Paris, out of Prague. You actually have a doable itinerary if you do that. Otherwise, you need to plan to fly to Paris from Prague, taking up a good part of a day and adding expense.

Also, I don't know that you have room in your itinerary for a few road trips to Ceska Rybna. You likely have time for one day-trip from Prague (likely 5-6 hours of round-trip travel). Or, stay a couple of nights in Ceska Rybna area - but you would need to sacrifice a couple of nights from elsewhere in the itinerary.

This could work, and would be a busy (and epic) trip:

Paris - 4 nights (arrival day plus 3 days)
[expect 6-7 hours travel]
Lauterbrunnen Valley, Switzerland -3 nights
Lucerne - 2 nights
[expect 5-6 hours travel]
Munich - 2 nights
Salzburg - 2 nights
Vienna - 3 nights
Prague -4 nights [with a full day-trip to Ceska Rybna]

You could slow down and save some money by cutting out Switzerland. Go straight from Paris to Munich and spend more time in Bavaria and Salzburg. You would still have a great German mountain-culture experience and cut out one of your long travel days.

Paris - 5 nights
[expect 5-6 hours travel]
Munich - 3 nights
Salzburg - 3 nights
(OR 4 in Bavaria, 2 in Salzburg and rent a car to see more of Bavaria and Alps - drop the car in Freilassing, Germany - metro-area bus ride to Salzburg, but avoid the cross-border drop fee)
Vienna - 3 nights
Ceska Rybna - 2 nights (one full day here)
Prague - 4 nights

You wrote about Paris:
"Recommendation on which area of Paris is central and easy accessible to train or bus? Hitting the major sites.....Eiffel Tower, Arch, maybe Versailles, historic cemetery, Notre dame and other suggestions? Perhaps an afternoon or evening boat trip."

We stayed on Ile de Saint Louis, very close to Notre Dame. It's also very expensive. Next time I'll probably stay in the Latin Quarter. The public transit is very good in Paris. If you search this forum you'l find threads where people suggest their favorite areas to stay (arrondissements). As far as other sites to recommend, what are your interests? I notice you don't have any museums on your list, but you might like Sainte-Chapelle which is near Notre Dame.

I'd probably pick three or four locations and give each one almost a week. Something like:
Paris-Switzerland-Austria or Paris-Austria-Prague.

Ruth, I think you are cheating yourself by visiting 5 countries in 21 days -- not 5 regions or cities in 2-3 countries but 5 COUNTRIES. You'll spend too much time commuting from one place to another without having much of a chance to EXPERIENCE the culture you are traveling so far to see.

Great post Ruth, stick with your original plan.