You are partially right, but while you are defending the rights of the mentally ill people, you should think about the sane persons too. It is not a single player game, if you with your mental illness ruins the game for others, then you are not fit to play this game. So if somebody has any kind of mental illness, and he runs down mid because he feels so, he deserves a punishment no matter what. I signed up to play the game and have fun, if he tries to sabotage me, i would like to see him banned. (Now i'm talking about gameplay related offenses and not chat.)
Nobody forces you to play the game. And NOT EVERYBODY is fit to play this game. Just like in real life, not everybody is fit for every kind of jobs, we do not have to think about disabilities, but even just physical differences, some people are stronger than others, so they can do jobs which requires that kind of skill, some others are more intelligent, so they will do jobs which fits them. If you are mentally ill, and your mental illness prevent you to play this game, and keep it playable for others, then you should not play the game. Go play single players, or other multiplayers, where your disability doesn't ruin the experience for others.
Because Joker's character is clearly mentally ill, because of the actions he makes. No sane person would do stuff like that. Usually the villains do stuff what no sane person would do, you need to be at least a psychopath to do stuff like that. Also, in a way Batman also have a mental illness, and i do not know too much about the comics, but if i know well, even he admits it, that not everything is correct in his head, and he do not want people to turn out the same as he is. (I might be wrong on this one.) And also the point of comics is to entertain, and you need some kind of villain there. The crimes committed by sane people are much more boring, than what a crazy villain would do, so they will have crazy villains for interesting plots.
Of course these words have negative connotations, because mental illnesses are negative.