Train travel across many countries (Germany, Czechia, Austria & Switzerland)

In Berlin you can choose any hotel near the stations between Zoologischer Garten and Warschauer Straße (see reduced network map). Train to Prague is cheap if you book enough in advance, check prices at Bahn website.

Basically start here: www.seat61.com

Everything you ever wanted to know about train travel and now do not have to ask...

First, I don't think you're allowing enough time for Berlin, unless you've been there before. I would suggest 3 nights bare minimum - four would be better, especially if it's your arrival city. In general, your plan may be too ambitious for the time you have.

In Berlin, look at Hotel Kastanienhof for a more authentic experience in Prenzlauer Berg. It's easily connected to mass transit via the tram that stops right outside.

In Prague, Hotel Hastal. Prague probably deserves four nights - 3 at a minimum. Check out the Living Prague web site and consider one of Jason's walking tours.

You may find Vienna a bit redundant after Berlin & Prague. If you want to vary your experiences, perhaps fewer nights there, or skip it. Others may disagree.

I liked the Hotel Julian in Prague. It was very comfortable. In Vienna, I stayed at the Pension Kraml near Mariahilfer Strasse, an arterial street. Kraml is a family-run budget hotel, but the room was big and quiet. The owners are very friendly. As far as Salzburg, I stayed there in my youth hosteling days.

Hotel Kastanienhof for a more authentic experience in Prenzlauer Berg

Info: Kastanienallee is pure tourism and gentrified area. What happens to the authentic people there you can see in this documentary about Kastanienallee. Maybe geo-fenced.

You have too many destinations for two weeks.
What are your priorities?
We always add a day to our first destination, because of the chance of delayed/ canceled flights since COVID. It's advisable to build in that cushion. Of my last three international flights, two were significantly late.
Switzerland is a two week trip in itself. Do you want to actually be IN the Alps or in the cities? I enjoy this country so much, that I recommend leaving it for another trip when you can spend all your time there.
Eurail passes have not been a great deal in recent years.
Perhaps review the RS Guide book itineraries, even the RS tour itineraries, do more research so that your interests and priorities can be more defined, then come back here.
Have fun planning!

Info: Kastanienallee is pure tourism and gentrified area.

Someone needs to notify Rick Steves:

▲▲ Prenzlauer Berg Lively, colorful neighborhood with hip cafés, restaurants, boutiques, and street life.

We were in Berlin 2 years ago, and we were somewhat disappointed in the city and their museums.

The train that goes to Prague goes through Dresden, and we absolutely loved the city. It was perhaps Europe's most beautiful city before the end of WWII, and the museums are fabulous. We suggest stopping there for 2 days.
Prague is another great city worthy of a 4 day visit. The train down to Vienna takes 4 hours (200 mi.)
Vienna is one of our favorite cities with great museums, music and elaborate palaces. It too is worthy of 4 days minimum visit.
From Vienna to Salzburg is 2.5 hours by train, and it's also one of everyone's favorite towns. It too is a music city.

For the time you have, it's doubtful that you'll have time for Switzerland. I found Zurich just a place to fly out of--an otherwise dull place. I'd take a train from Salzburg to the Munich Airport and there are a bunch of U.S. cities you can fly to non-stop from there.

I have read on forums that Lucerne is a good 'central' location/base to stay in Switzerland and then we can do 'day' train trips to various places around Switzerland. Is that the best route or city to stay in for good access?

Luzern (Swiss spelling) is a good base for exploring Luzern on day trips. You can explore the mountains nearby — Pilatus, Rigi — and the lake itself. Lots of starter info here https://www.ricksteves.com/watch-read-listen/read/articles/luzern-swiss-mix-of-urban-delights. With all that you have planned including Berlin, Prague, and Vienna, you’ll only have a few nights here (or anywhere) and Luzern has enough to keep you very busy for at least 3 full days.

FWIW, friends of mine just spent 18 nights on a European holiday, hitting Scotland (2 locations), Switzerland (also 2 locations), and Sweden. It was an insane itinerary, their first big trip after retirement. They acknowledged upon returning home that they tried to do too much. They lost 2 full days to being sick with COVID in Scotland and another very full day getting to Switzerland from Scotland, and then lost a day to extreme exhaustion upon arriving in Zurich. They only extended stay they had where they could relax a little bit was when they got to Lauterbrunnen and settled in for 5 nights.

Sometimes — usually in my experience — less is more. Truer every year!

Looked @ the 15 day Eurail unlimited pass which is $500 per person, is
that the more economical pass to buy or better to buy daily train
tickets a few days before?

Neither. The economical way that you buy tickets in Europe is to buy your tickets from City A to City B online through the actual train company as far in advance as possible. It is priced like airline tickets. The longer you wait, the more expensive the tickets become. Buying a few days before will be double or triple the price you would have paid 4-5 months in advance. Eurail Passes are a luxury and never save anyone any money. Eurail doesn't even operate any trains.

Berlin to Prague - 18.90 Euro per person. https://www.seat61.com/trains-and-routes/berlin-to-prague-by-train.htm

Prague to Vienna - 14.90 Euro per person. https://www.seat61.com/trains-and-routes/prague-to-vienna-by-train.htm

Vienna to Salzburg - 19.90 Euro per person. https://www.seat61.com/trains-and-routes/vienna-to-salzburg-by-train.htm

Salzburg to Zurich - 29.00 Euro per person. https://www.seat61.com/international-trains/trains-from-Salzburg.htm#Salzburg-Switzerland

Total cost - 82.70 Euro. Sounds a lot better than 500 Euro.