I did a similar European gay-advenutre-on-a-budget back in 2004 and had the time of my life. Hope you have an awesome experience as well!
Berlin: I've stayed several times at the Circus Hostel on Weinbergsweg in Mitte. Very cool area, and this is almost universally praised as the best-quality hostel in Berlin. A pleasant walk to the Prenzlauer Berg bars, and close to the U2 for trips to the Nollendorf Platz bars.
Barcelona: Have stayed in both the Gothic Point hostel and one in Raval who's name currently escapes me (perhaps on c/- Sant Pau?). Both were good quality, and the latter (unnamed) one is an easy 15 min stroll to the bars in Eixample.
In Paris, you want to be in the 3rd or 4th arrondissements to be close to the action.
Have fun!