I don't watch much horror anime, but
Boogiepop Phantom and
Serial Experiments Lain both have horror-esque qualities, but are really more sci-fi thrillers. Westerns...ummmm...there ain't much. The best you could hope for are titles that have western qualities like
Cowboy Bebop and
Trigun. Sci-fi, on the other hand, is quite plentiful. Cowboy Bebop again would be a great choice as would be the Robotech/Macross discussed earlier.
Planates and
Space Brothers for hard sci-fi (like the films Moon or Sunshine)
Neon Genesis Evangelion and
RahXephon if you'd like some psychological thrown in with your mecha. Ghost in the Shell (
movies and
TV could also fit into your last category as it often involves solving crimes as well. Really, for sci-fi, there is a ridiculous amount of great material to choose from. And specifically for crime drama, I would recommend
Monster with the caveat that it is very long and certainly a lot more involved than the general plotline of a doctor tracking down a serial killer.
Other titles I consider at the top of my list: anything by
Satohi Kon (if you're at all a fan of David Lynch's work, I highly recommend).
Kino's Journey and
Haibane Renmei which are pretty difficult to associate with anything else; they're just that unique. If you want depressing:
Grave of the Fireflies,
Now and Then, Here and There, and
Gunslinger Girl (actually, this one is more creepy than outright depressing).
Revolutionary Girl Utena and
Penguindrum, both of which are from the same director and both start off rather frivolous but get rather depressing as they go along; they're also rather...ummmm, "fanciful." And, just for the hell of it, a period romance:
Emma.