I've not seen most of these. I have difficulty finding them with English subtitles, but I hope that will change soon, as they're all in my To Watch list.
Falling High School Girl and Irresponsible Teacher, deep depression tending to suicide (I have seen this. It does have a teacher entering a relationship with a student, which is generally very problematic with me, but it's handled well here. The teen is in charge and more mature, in many ways, than the teacher. They're both lonely, desperate people.)
He Who Can't Marry, OCD, mental illness
Iyashi no Otonari-san ni wa Himitsu ga Aru, mental illness, obsessiveness
Kururi, amnesia
Marry Me!, agoraphobia (I enjoyed this quiet, sweet romance.)
Nodame Cantabile, aviophobia, mental illness
Roppongi Class, psychopath
Sachiiro no One Room, depression
