Switzerland-Salzburg-Rome itinerary

You don't want to look for Raileurope. They charge too much. Use trainline.eu instead. They have most of the trains you want, and if not go to the country's own train site, such as sbb.ch/en for Switz., or http://www.oebb.at/en/ for Austria.

I'm quite sure there is not a non-stop Interlaken to Salzburg. There was and maybe still is a non-stop overnight from Salzburg to Zurich, and that's still 3 hrs short of Interlaken.

I'd be amazed if there was a daytime nonstop Salzburg to Rome. You might check out flights via skyscanner.com

You shouldn't have to rent a car. If there's a place people want to visit, Europeans almost always have a train/bus way to get there. They do this MUCH better than we do in the US.

Your travel plans seem to backtrack: south then north then south again. And, Salzburg to Rome is a LOOONG way. We once took an overnight train from Salzburg to Zurich and that was 7 hrs as I recall. Going to Rome would be twice that!
Once you are south at Interlaken, you might consider trains to Italy via Spiez, or, as I mentioned before, discount Europe flights should be considered. (Two common ones are Ryanair.com and Easyjet.com)

what is the exact website for Euro rail - to book tickets and check train timings

No such website. Eurail is the name of a company in North America that sell passes for use on the various railway companies in Europe. Rail Europe is another North American company that resells passes and sells some tickets often with a markup.
There are dozens and dozens of railway companies in Europe. Mostly national ones, but also local ones and in some cases (but rarely) two different competing companies on the same route.
If you want to know train times, the best place is the DB (German railways) website, this has data for most of Europe: https://www.bahn.com/en/view/index.shtml
Buying tickets, start with the company actually running the trains. There are some Europe-based websites which sell at the same price as the rail company (trainline.eu loco2), but they only cover some countries.
I suggest you start by reading the "Man in Seat 61" website, which is a consumer website explaining how train travel works and where to get info and buy tickets for each country: https://www.seat61.com/

is there a non-stop train from Interlaken to Salzburg?

No, change in Zürich HB. But trains (unlike planes) are designed to be an interconnection network with interchanges. There is minimal time loss changing trains (unlike planes). Look up times on the above-mentioned DB site.

for Salzburg to Rome, is there a non-stop day train? The only non-stop I found was an overnight one.

As above. All trains have intermediate stops, including the overnight one. Do not confuse a "stop", where some of the other passengers get on and off, but you stay in your seat; with changing trains, where you have to get off one train and on another.

would we require to rent a car anywhere? I know inter-city train system is good. But what about visiting places within a town/city?

Within a city/town is exactly where you do not want a car. Cars are kept away from city/town centres. You either can't drive there or there is no parking. Use public transport.

No comment about your itinerary, well planned.
but, could you fly "open jaw"? Out to Zürich and back from Rome. That would save time.