Do PhD programs care about the personalities of applicants? Can a very extroverted person, for example, be more favored than a very quiet person? Or is this not ethical? Can someone who is psychopathic be barred from admission?
Yes, decisions based at least partially on personality do occur.  It's not clear to me what the legal technicalities are - for example, depending on your country there may be laws regarding things like discrimination against the mentally ill - but in practice an admissions committee can do whatever it likes, and nobody will be able to prove there was any illegal discrimination.

The way it typically works is that having a particularly pleasant or agreeable personality won't help you, but having an unpleasant personality may hurt you.  If you seem likely to be difficult to get along with, rude, disruptive, uncooperative, or otherwise problematic, then that will generally be held against you.  (This may be judged based on your personal statement, letters of recommendation, etc.)  It won't necessarily doom your chances: some committee members just won't care, and others may be willing to excuse bad behavior if you are sufficiently talented.  However, on average it will hurt your chances, sometimes substantially.