Yes, my Kurzweil K2000 (http://en.wikipedia.org/wiki/Kurzweil_K2000) supports fully customisable temperaments – you can even tune it to the Bohlen-Pierce (http://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale) scale if you want!

That's a sweet, sweet synth.

Now I'm kicking myself for selling my gear. I had an Alesis Ion. So much fun.

Workstation-style keyboards do (the ones that have a synthesizer and a built-in sequencer as well as other features). This is implemented using the MIDI tuning Standard which has been around for years. Such functionality is usually not available on simpler synthesizer, which allow adjustment of concert pitch but not much else. There just isn't much demand for this outside of professional composition circles, for the same reason that Barbershop Quartet music (which relies very heavily on just intonation for its ringing harmonies) is an extremely niche musical taste.

It's useful if you want to write non-western music, eg Arabic or Indonesian or Indian styles which have different scalar intervals from Equal Temperment, but only if you're very concerned with authenticity. The fact is that if you're composing for a Western-music audience you can get most of the 'feel' of other musical traditions by selecting a suitable scale and playing in equal temperment. If you're composing within the musical tradition of a non-western culture you're less likely to be using something conceptually oriented around a piano keyboard in the first place. In a lot of musical traditions (and indeed country and folk music in the west) the emphasis is not so much on tonality and composition as having a good repertoire of stock musical phrases that everyone is familiar with, and virtuosity consists in skillful improvisation employing those musical tropes.

Some, yes. (Generally those geared more toward the synthesizer/MIDI crowd.)

Of course if your synthesizer is your computer, it's a question of what the synthesizer software does, not the keyboard; and the answer is the same.

Not many of them. They often let you change the fine pitch of the whole keyboard though (in old school analog keyboards this was because the whole instrument could go out of tune over time / as it warmed up I believe, but these days it's sometimes still present, maybe as a way to play with out of tune instruments). According to other responders, some keyboards do, but it is not standard.

But besides using the built-in sounds in your keyboard, you can use most modern keyboards to send MIDI notes to some external (or virtual) synthesizer to play.

I just looked into handling MIDI for different temperaments. MIDI is a way to digitally represent each note as an integer (so let's pretend Middle C is the number 80, the B directly below would be 79, etc). The MIDI standard itself afaik has no support for other temperaments, but the receiving synthesizer can. So some VST instruments (virtual instruments you run on your computer for either recording or live performance) do support other temperaments, but it's not across the board. If you want to write or perform in a different temperament, you probably are limited to using VST instruments that support it.

If you're sending MIDI to a synthesizer that doesn't support choosing temperaments you can mostly fake it by sending pitch bend signals.

Many of them do. Sometimes you can select from a menu of pre-defined temperaments, sometimes you can individually tune each pitch class (e.g. you can tune all the G's 4 cents sharp if you want), and sometimes you can tune all 88 (or so) notes individually.

In general, you run into problems if you want more than 12 notes per octave because a) standard keyboards are really inconvenient for that and b) midi wasn't designed to support microtonal music in any consistent way, so you have to employ strange kludges, like use a separate channel for every pitch class and use pitch bend for tuning.

My Roland digital piano has several temperaments it can use including meantone, just, pythagorean. It's a low-midrange model HP203, make to look and sound more like a piano.

Some high-end models do