hopefully they'll still be in the hunt for a 19th league win... silly me of course they will, we're not talking about Liverpool
Cheeky monkey! I have two words for you: Chel-Sea!
Team allegiances aside, I hope you do get to see a match, the derby would be even better.
If you don't want to visit France, then there are plenty of flights to be had from London (avoid Ryanair and don't forget to check BMI and British Airways as they often have good deals to certain cities) so your other destinations, so you don't have to go to Paris if you're not interested. If, however, that's the only French city you want to visit, then I would suggest the Eurostar as it's a great ride and quicker than flying to Paris, just book well ahead for good deals.
After that, you can certainly include the other cities you've mentioned by flying between them rather than taking long/overnight train trips. As an aside, the Rome-Milan trip takes only 4ish hours on the Eurostar (Italia) and would be a better option between these two cities than flying as the Milan airports (there are 4!) are a fair way from the city itself. On Milan, I'm not sure I'd be including it in an itinerary as it's the least "Italian" of major Italian cities. Of course, if it's all about the football, then by all means visit! ;D
As a suggestion, and based on the cities listed, you could consider the following:
(from London or Paris, fly to Berlin and begin from there. Check Air Berlin for flights)
Berlin - 3 full days in the city.
train to Prague via Dresden for a few hours/lunch stop. Dresden is a compact and interesting city, so it's a good side trip to take between the two.
Prague - 3 full days; this will allow two days of sightseeing and wandering and pubbing and a day to do a day trip to Kutna Hora (aka the "Bone Town"). If you want to do a little more sightseeing of Czech Republic, then you could travel from Prague to Cesky Krumlov and stay the night before continuing on to Vienna. It's a little bit of backtracking by train to get from there to Vienna, but it's a pretty city (World Heritage Listed) and worth a stop if you can include it in your itinerary. The train ride from Prague to CK is around 4hrs and then from CK to Vienna is 5hr30mins. Whereas direct Prague-Vienna is 4hrs30. Up to you, but do a little research to see if Cesky Krumlov is of interest.
Vienna - 2 full days would be good hear, depending on your interests.
I note
Munich isn't on your list, but I think you may enjoy it as it's a great town with day trips to be had in the region (half day to Dachau, full day to the castles like Neuschwanstein). If you do include it, allow 2 full days depending on the day trips you want to make. Train from Vienna will take around 4hrs30.
Salzburg 1 full day is fine for Salzburg, unless you plan to do a day trip to the Salt Mines (Salzkammergut), Berchtesgaden or even a Sound of Music tour (!), then you'll need an extra day. A train from Munich is only around 2ish hours.
Another city you could include is
Venice. Depending on your interests, 1-2 full days would be good. The train Salzburg to Venice isn't short (around 6hrs) but the scenery is spectacular. Failing that, you could always go Vienna-Salzburg-Munich then fly Munich to Venice instead (Lufthansa sometimes has good prices on this route if you book well in advance).
Train from Venice to Rome is 4hrs on Eurostar (Italia), but you could always take the train to Florence, spend 2-3 full days there and finish in Rome to fly home from there. Florence, if you haven't been, is a great city for sightseeing as well as eating. OK, I'm biased, but if you can fit it in then go for it. You could always to a half-day trip to Pisa to look at some leaning thing too. Otherwise, there are places like Cinque Terre, Siena, Assisi, depending on the time of year and your interests.
Rome - 3 full days would be a minimum for Rome, and if you're planning a (long) day trip to Pompei then add one more. As mentioned, look for open-jaw tickets to allow you to fly in to Manchester but out of Rome so you don't have to back track.
That should give you a basis for your itinerary and you can tweak it with time/cities according to your interests and budget. All of the cities mentioned have good nightlife, just in different forms to what you may be used to, so once you've got more of an idea on your itinerary, we'll be able to fill in those gaps.