An anime's soundtrack is usually made specific for the show. The same cannot always be said about the anime's opening or ending themes, though.
When it comes to producing an anime, multiple companies work together to make a show happen; these multiple companies make up what's called a Production Committee. Depending upon the show, a show's production committee will include a music label (such as Aniplex or Lantis), and this label can provide a composer to make music for the anime. There are cases and times, though, where a specific composer can be brought in from a certain label (such as Yuki Kajiura being brought in to work on Madoka Magicka), if someone in the production staff has a connection or something like that.
A computer can synthesize a lot of instruments pretty dang well, and it's assumedly cheaper to hash out a soundtrack through computer software rather than finding an orchestra/band, teaching them the music, and then recording the performance. Using live instruments for anime isn't unheard of (such as for higher-production films), but I couldn't tell you a percentage of how much certain anime shows use what. For vocal stuff, I don't specifically know how that gets managed, and I wouldn't want to lead you astray by even more conjecturing.
Finally, to answer your question about how they decide the feel of the sound, I'm led to believe that it's hashed out at meetings during an anime's creation (usually a bit closer to the end stage of things), with the composer (or someone representing his/her label) talking with the anime's production staff. Words could be thrown around that depict a certain feeling, or a certain genre of music or something to go for. If the production staff knows more about music, more specific terms may end up being used.
When it comes to how the music itself is created, and how composers get the training to get their jobs, just look up "music composition" and "digital music composition" on Google and it'll start to point you in that direction. A lot of the same general composition and creation processes apply, although I can't really tell you if they do something specifically unique in Japan composition-wise. Anyway, people can go to college to study music theory and music composition, and from there, they may choose to work specifically for anime, or choose to work anywhere else (like composing for live-action TV, or for movies, or commercials, or for the entertainment industry, for example). It's a career and profession like any other.
Sources:
http://www.animenewsnetwork.com/feature/2012-03-05 - Justin Sevaki's "The Anime Economy" series. Doesn't really delve into the specifics of how an anime is made, but it explains the production committee bit fairly well.
http://web.archive.org/web/20081002032241/http://gabrielarobin.com/279/newtype-yoko-kanno-and-shoji-kawamori-macross-f-ost-1-interview-translation - An interview with the chief director and composer for Macross F (provided through the Wayback Machine). The translation isn't the easiest to follow, but it provides some information about how the music came to be (Macross F also had a lot of vocal music too!)
Interview contained in the DVD release of the 2nd half of Zaion: I Wish You Were Here. Show wasn't that memorable, but it had some great music! The music composer for Zaion was contacted and brought in while the show was still in its planning stages and they talked about what feel to attempt for the series. However, that being said, Zaion may have been a bit different because it was made for Internet distribution, an unheard of thing at the time.
