Justify if the narrator has a mental disorder. (He is always trying to convince himself that he is not mad and explaining his actions.)
Find clues about if the narrator should be taken to the Jail (because he committed murderer or a psychiatric hospital)
I need arguments about why (I am defending the point of view of the psychiatric hospital).
You assume he's trying to justify his actions to himself, but a good question to begin with is whether he has a different implied audience. The way he speaks, I think the audience is not him. Consider the first sentence: "TRUE! --nervous --very, very dreadfully nervous I had been and am; but why will you say that I am mad?" He says "you" here. If he were talking to himself, wouldn't he say something more like, "But does that really mean I'm mad?"
At the end of the first paragraph, he says, "Hearken! and observe how healthily --how calmly I can tell you the whole story." He's clearly talking to someone else.
The next question is who that is. One possible explanation is that he's talking either to the police who arrested him after he outed himself at the end of the story as his wife's murderer. Another possibility is that he's talking to his lawyer. Yet another is that he's trying to convince a psychiatrist, or even a jury or judge.
Come to think of it, he needs professional help no matter who the audience is. I mean, if he is truly talking to himself, he definitely needs professional help, since he's having a full-blown conversation with himself as a distinct other person (in this instance, I picture him already sitting in a padded room in a strait jacket).
If he's trying to convince someone else, his nervous defensiveness about his mental stability (such as, "The disease had sharpened my senses --not destroyed --not dulled them." Also...what disease?), his ravings about how calm and balanced he is (when he clearly is not, not to mention the actions he admits he took (how and why he set about murdering an innocent man) all point towards absolute madness.