In modern day language, what is a Halachic Shoteh?
In other words, is someone who is schizophrenic, clinically depressed, OCD, etc. free from Mitzvos?
Very similar to the Downs question.
Rambam says you have to see each case in-person to call these shots.
Basically, if someone lacks the consciousness or control to perform a given mitzvah, they're not obligated in it. (This is called "a shoteh for one thing.") It's possible for someone to, G-d forbid, have an eating disorder such that there is really no way they could say, eat a piece of matza on Pesach; at which point they're not obligated. (Though hopefully the person can be rehabilitated.)
While Rambam says in his Laws of Testimony that "a person who acts crazy with regards to one matter but is normal in all others" is considered a Shoteh, it appears the majority opinion since the 1700s has been that this was limited to the definition vis-a-vis qualification to serve as a witness on ritual matters. With regards to mitzva obligations and halachic competence, each action is its own question.
The classic, "full-spectrum shoteh" described by the Talmud is exempt from all mitzvahs. Such a person repeatedly does actions, without reasonable explanation, that indicate he is totally not conscious of reality around him, these are actions no normal person would do. Gemara talks about someone who tears their clothes (but is not an absent-mind professor), goes out at night when it's incredibly dangerous to do so (implying he doesn't realize it's night time or it's dangerous; if he explains that he has a fever and needs to cool off, or even that he's having a panic attack and needs to get out, that mitigates), and/or sleeps in a graveyard (yuck! Must be he's not conscious of reality. Unless he explains he knows exactly what's going on but this is a religious ritual of his.) Rambam gives additional examples such as someone who goes around naked or throws stones -- again, because he's not conscious of reality, or not in control. If there's a reasonable explanation for these behaviors that's a different story.
Dr. Rael Strous proposed in a 2001 paper that the "classic Shoteh" of the Gemara was probably some type of schizophrenic.
Of course, any condition that's currently under control of therapy and/or medication, is a moot point. If someone is sane when the moon waxes and Shoteh when it wanes, he's fully halachically normal when the moon waxes (called "itim chalim itim shoteh"), so too if it depends on whether he's taking his pills.
Low functional IQ: definitely not a classic Shoteh. Obligation in any given mitzvah would depend on their mental capacity for it, but usually applies.
Clinically depressed: the Aruch HaShulchan described this case. (From his description you can tell he was a pulpit rabbi who had seen it.) A serious cause for all kinds of halachic concern, but not a Shoteh. From Rabbi Torcyner's sourcesheet:
(E.H. 121:13) Those who have melancholia, who do no crazy actions, just sit in sadness and are distant from human companionship and have no interest in eating - but if given, they'll eat; they don't initiate conversation, but if asked, they answer a bit; they don't talk crazy either, it should be contemplated if they are considered a Shoteh or not. Logically this lacks signs of Shoteh, and is just a disease of sadness. It depends on the actions and motivations as seen and understood by the Beis Din (rabbinic court) (Rashi Hagiga 3 אני שמעתי חולי האוחז מתוך דאגה).
OCD: most likely there will be some mitzvahs they can't handle right now, and others they can.
Bipolar/hypomanic: in Camden about 60 years ago, a man would go into fits when he thought he was super-rich and would spend like crazy. The local posek (don't recall his name) said that such behavior doesn't rise to the level described by the Gemara (plenty of people overspend), so he wasn't a full Shoteh. Rabbi Yosef Eliyahu Henkin concurred.
I personally know of a case where a man was convinced he is Moshiach, and his wife wanted a divorce and most Rabbis said that he is a Shoteh and can not give a divorce. However after HaRav Moshe Feinstein Zatzal met him he said that he is not a Shoteh and may give a divorce, and gave a letter to the lady that he determined that her previous husband was not a Shoteh.