Please do not apologize for "ranting' here, that is what we are here for, to give you some comfort and some support and help you get through these rough times, I am so happy you have reached out to get some of this stuff off your chest. Also I get it, the facial hair and the hair loss, it is all relative to you and you have a right to feel sad or feel how you do about it, just because people "out there suffering a lot more severe things and I should be grateful"..if it is bothering you it is legitimate and does not make your problems any less than anything anyone else is going through.
OK so the girl got upset as she felt you didn't consider her or that she had a cat, can I say I have 3 and they all get along perfectly well and love each other, I am not sure why she felt she was the only one that was allowed to have a pet, that your pet would effect hers? I think that perhaps there are two ways to handle this girl, the first being, let it go..let her be the one who is all angry and let her be the one with the issue, maybe in time you can work it out and tell her that you are sorry you upset her, the kitten was a gift and if she cannot talk it out to resolve the issue then it is on her. That way you can move on with your life and if she wants to ignore you that is up to her. The other option is to ask her again if you can sit down with her and chat it out, that you do not like having this situation and that it is actually impacting other people in the friendship group. If you show maturity with this hopefully she will follow, however at the end of the day you cannot make this girl talk to you or "get over it"..sometimes we have to accept the situation and just work around it. I know this is not helpful but that is sometimes how the cookie crumbles..to quote a cooking phrase!
Perhaps you have to make plans with your friend outside of her house so as you don't have to feel excluded from the friendship and so you can still see your friend without having the girl and the house be an issue.
ERRKK...this is so horrible and it really is tough being a young person these days. I am so sorry you are going through this, huge hugs to you.