Route sounds definitely doable but on most of the routes there are no overnight trains. You can use them from Rome to Venice, Venice to Salzburg and Budapest to Prague. The other trips aren’t that long, so you can easily use a daytime train.
Rome to Venice by high-speed train is also possible and doesn’t take too long.
Venice to Salzburg during day takes long but scenry is probably one of the best during your trip (beside the Interlaken to Milano trip on the way to Florence).

We have just done a slightly less rigorous journey but with the same style.

We found treating the journey as a “travel day” and then staying 2 or 3 nights was about the right mix. We tried to find hotels/apts near the central station so we could drop off luggage and then have a gentle walk or bus to the centre for the evening.

Our journeys varied from 3 to 10 hours.

We stayed in Siracuse, Salerno, Bologna, Nuremberg and then an overnight in Brussels before E* home.

You could also stay longer in some places and then day trip to nearby (1-2- hours each way) places.

We did Nuremberg for 3 nights and covered three local towns/cities using regional trains. Venice for example is easy reach of Verona, Lake Garda and even Bologna (but remember res fees for fast trains in Italy. Vienna is good for day trips to Linz, Graz, Munich , Bratislava and even Budapest. We are there next week and staying in a modern apartment block right by the station for well under £100 a night.

If staying in Venice I suggest staying near Venice Mestre station and travelling in to Venice either by tram or train for a couple of Euro - saves a fortune and hotels much better than the tourist traps in old Venice. The Venice public transport ticket seems expensive but is essential for water bus transport - individual journeys are extortionate and the pass covers all transport in the Venice region - including across the lagoon.

We only used 5 travel days and a couple of days paying local trains so 15 will be more than enough.

First Class is well worth it for the relative peace and space on most trains. You can also use the lounges in Austria and UK (Kings Cross definite) and you even get a small meal and drinks on Eurostar in Standard Premier.

Also remember you can use your pass to get to and from London on the same day as you take Eurostar.

We considered night trains but for a civilised cabin (not a couchette) are difficult to find and extortionately expensive even with a E* pass because fees are per person.

We never paid over £100 a night and daytime journeys, even long ones, have the advantage of seeing the scenery. We took trains at times that allowed a civilised breakfast, arrived at reasonable times and bought and read “Europe by Rail” before we left. This was well worth the cost  as it highlights the key elements and history of your route.

All these sectors are far too short to warrant overnite-triptimes of 2/3, maybe 4 hrs are not fit for that.

THis also means that you do not loose too much time: start a bit early, arr at lunch-perhaps drop even luggage at HTL already, or in locker.

IF you want to use overnite (IMHO much overvalued-you will not sleep very well and have the sametime needed to recover) you need to re-arrange-easy on a pass-and perhaps doubletrack.

(f.e. Roma-Wien-Venezia-München). A real sleeper place in a NightJet is in fact not that much cheaper-if at all-as half a double in a budget HTLchain like Ibis or HotelBB.