Planning for a Europe trip in spring 2018

Cut out Germany and Poland since none of your must sees are in those countries
This works I think:
Fly into Rome out of Prague or vice versa:
Rome to Florence to Venice to Salzburg to Vienna to Budapest to Cesky to Prague

How many nights in each city would be enough to see the highlights?

If you have a full 3 weeks then ...
Fly in to Berlin - 4 nights
Train to Prague - 3 nights
Bus to Český Krumlov - 2 nights
Bus to Linz then train to Budapest - 4 nights
Train to Vienna - 3 nights
Train to Salzburg - 2 nights
Train to Munich - 3 nights
Fly out of Munich

If you only have 2 weeks then stick to just Italy.

Rome needs three full days, Florence needs two full days so that's really 7 nights remember .. Salzburg needs one.. ( sorry I think its skippable , but that's personal tastes)

I don't believe in less than two nights in one place.. because two nights is ONLY one full day and I think if isn't worth at least one full day then its not really that high on your list of must sees, so delete it. Your trip is eaten up fast once you actually sit down and map out the NIGHTs.

Remember the more you move the more it costs ( even with cheap flights and train tickets) and that each day you move you have eaten into actual sightseeing time, so the fewer moves to places means you really get to see the places you do go to..

I also think for what one spends on airfare to Europe, and the hassle of getting time off work ,that if you go , go as long as you can, its not like most of us can go every year or for some even every decade.. so go for the 21 days if you possibly can.

Rick's "Europe Through the Backdoor" and his planning tips are a good start:
https://www.ricksteves.com/travel-tips/trip-planning/itinerary-tips

Minimize one-night stands. Even the speediest itinerary should be a
series of two-night stands.

Leave some slack in your itinerary. Don’t schedule yourself too
tightly (a common tendency).

Assume you will return. This “General MacArthur” approach is a key to
touristic happiness. You can’t cover all of Europe in one trip — don’t
even try. Enjoy what you’re seeing. Forget what you won’t get to on
this trip.

My own bias is more time in fewer places and I like to start slow to recover from time travel. Ten cities and five countries seems like a lot of travel for two or three weeks requiring a lot of transportation planning.

Have you read the section of this website under "travel tip" then trip planning? that list is way too big for 21 days, but it sounds to me like you have already chosen your must sees. There is advice about how to choose which direction. Is this your first trip to Europe? Weather is one consideration - the other is going from least cultural shock to greatest cultural difference. I'm not sure what that would be in your case.

Sorry, but your list is far too long even with 3 weeks. Allow on average 6/7 hours door for each flight that you take and as you have some big hitters such as Rome and Venice, base your trip on two destinations a week.

Easter is early next year, so schools and universities will be taking holidays the first 2-3 weeks of April. Rome will be busy and expensive at Easter. Some would spend Easter in Rome, whilst others would say avoid it, as it's too busy.

Fly open jaw. There is no "right direction " to travel in, but it makes sense to cover all of Italy at the same time. Look at rome2rio.com for route options. The order should be worked around your international flights.

Nights
1 Arrive in Prague about 1 pm jet lagged
2 See Prague
3 See Prague
4 Morning: Prague to Cesky Krumlov on Bean or CK Shuttle
5 Afternoon: Cesky Krumlov to Vienna on Bean or CK Shuttle
6 Vienna
7 Vienna
8 Morning: Vienna to Gyor by train - taxi to Archabbey at Pannonhalma
9 Morning; tour Gyor. Afternoon: Gyor to Budapest by train
10 Budapest
11 Budapest
12 Budapest & Szentendre
13 Budapest
14 Wizzair has some discount flights for under $100 to Italy (But I would stay in the east and visit Bulgaria, Romania or Ukraine.)

If your trip is 14 days, then Italy will eat up most of that time, however if you have 21 days, take in Munich/Prague/ perhaps Vienna/Budapest.

Some say you can see Rome in three days, I have been to Rome twice, both times for a week and I still have things to see there.
Venice is good for two days, Florence for 3-4.

Rome 4 days
Florence 3 days
Venice 3 days
Salzburg 2
Vienna 2
Budapest 3
Czesky 2
Prague 3

Not just 1 night stands but eliminate 2 night stands. Unless travel
for you is a mere check-off list which 2 night stands illustrate, stay
for a minimum 3 nights per hotel.

MsEB, absolutely and, not always. We are fortunate enough to travel several times a year, so I have started to include a 1 or 2 night stay some place unique just to test it. I call them bonsai trips. If I like it we go for a longer stay on another tip. If we didn’t like it much, we know we prevented wasting 5 days on a dud. Having so many discount flights under $100 in and out of Budapest helps to make it easy to do as well.

The other thing I do from time to time is have a Plan B waiting. Plan B is when Plan A isn't all that I had hoped. Our first trip to Budapest, Budapest was the Plan B. After a few uninterested days in Vienna I was harassed by one too many of those street corner Mozart characters and we checked out and headed south. Best decision of a life time.