To what extent are the mentally disabled exempt from mitzvot, and what mitzvot are they exempt from? At what level of mental disability does this exemption kick in if it exists, and is it gradual depending on the level of mental handicap, or is it all at once when a certain threshold is reached?
Generally, they are obligated in whatever mitzvot their consciousness can support. If someone's actions indicate they're completely disconnected from reality (hard to define), then they could reach the threshold of a shoteh and be exempted from everything.
There are 50 places in Rambam's code where he takes for granted that a Shoteh (insane individual) is exempt from all mitzvot. He then writes that a Shoteh cannot serve as a witness, "as he is not bound by mitzvot." Additionally, "the exceedingly simple who cannot determine that two concepts are mutually exclusive" are disqualified from serving as witnesses, "and the exact gradation can't be described in writing, each rabbi has to make his own determination."
Rabbi Moshe Feinstein, like many others, read Rambam as saying that the "exceedingly simple" is not a Shoteh per se, but that there is a mental-fitness disqualification for witness eligibility, even if one is obligated in some other mitzvot. (Some suggest that such testimony never bears the risk of perjury, as he could always explain, "I didn't understand what I was saying." Rabbi Feinstein suggests that the Torah requires a witness to have a consciousness that supports all 613 mitzvot.)
There's a responsum of ... the Shevet Sofer, I think? about committing an individual to an institution that will help increase his cognitive function. The issues involve the availability of kosher food, and his ability to leave for his bar mitzva. Rabbi JD Bleich (in a yutorah mp3, and a J. Halacha & Contemporary Society article) observes from the language that the individual was developmentally challenged, not insane.