Personally, if I were doing it, I'd limit myself to just two cities: travelling between centres is tiring and time-consuming, and you run the risk of falling into that well-known syndrome of "if this is Tuesday we must be in Belgium" (as it were).

If you were to limit yourself to two, I'd say London and Paris. Whether two or three, I think it's reasonable to split them roughly equally.

As to St Patrick's Day, it's big in London (lots of Irish people and people of Irish ancestry), much less so in either Paris or Amsterdam. (Note though that most people in London do not celebrate St Patrick's Day or mark it in any way, so you need to pick your spot, obviously.)