I wouldn't mind universe crossovers as long as it doesn't affect either the canon and/or it is a comic episode. In fact I would pretty much love them if they are done right. I've never seen AvP because I heard it is awful, and because when I read the premise I didn't like it, so I count it as a failure to do it right.
For Alien vs. Star Trek: In Alien we were made to believe that the alien is the first extraterrestrial creature we meet, although it was far from the main focus, I believe it is important for fully appreciating it. As far as I recall, the events in Alien happen after First Contact, so you can't do a serious crossover, and I'm not certain how a comic one would work. Also the Borg were similar to the alien in many ways, only watered down for watching with the family (e.g. First Contact could have been a horror film if they didn't intentionally kill the tension every now and then).
A crossover with something like Doctor Who might work, although again it would have to be a comic one (because the two stories don't fit) and more Doctor Who-oriented (because Star Trek is too serious and pretentious).
The Star Trek universe has already been connected with others, for example Greek myths (like Who Mourns for Adonais?), Sherlock Holmes (Elementary My Dear Data, etc.), and it could easily adopt more modern stories in a similar way if rights issues can be resolved (which is why I believe copyrights should expire faster). It would be a cool thing, but it requires new series, because I would be disappointed if they waste a film for anything but an original in-universe story.