FAQ for
EuroTrip (2004) More at IMDbPro »

No. Matt Damon simply mimes the song. The voice you hear singing is the lead singer's of the band Lustra who wrote and performed the song for the film.

To put it simply, there are two new scenes in the unrated version with a lot of female nudity that have been cut for the R-Rated release of the movie. A detailed comparison with pictures can be found here.

(Correction: the above link is for the movie Road Trip, not Eurotrip.)

Unknown. If she was, she certainly wasn't by the end of the film.

Like any other recent teen comedy both R-rated and Unrated versions were released on DVD. The latter features a slightly longer cut showing some more nudity and profanity but also some prolonged plot elements. A detailed comparison between both versions can be found here.