13 is generally too young to be diagnosed with a personality disorder, and depending on how developed you are puberty-wise, sex might still seem icky at that age. Some people are just asexual, and never want to have sex, or only masturbate, and that's ok. The fact that you do connect with some people (your friends) and that the cashier scares you makes me think social anxiety rather than schizoid.
I used to be scared of ordering in restaurants for a long time, and phone calls to people I don't know are still scary enough that I tend to avoid them. A friend of mine who has social anxiety is fine in small groups, but he's introverted and absolutely hates parties. Autism spectrum problems can also cause problems with getting the rules of social interaction. It's not a good idea to self-diagnose, especially when so many symptoms overlap. You might want to talk to your school counselor about how you're feeling, since these things are interfering with your life. I saw you mentioned that your mother is bipolar, which could lead to some dysfunction that's not a personality disorder.
Honestly, I think you are just not a very socialable person which is okay. See my uncle, my mother, and my dad are skitsofrantic. I believe I am too, but I can tell the difference from my head trips most of the ime. See my uncle and mom are severely skitso, and being skitso does noyt only make you antisocial, you see or hear people all day everyday, and my uncle talks to them, even swings at them. And looking at him do this....he is swinging at mid air. I tend to see people but I try to sort the reality from the not,. Like someone being under my bed that is no more than 2 inches high is alittle not realistic. Also, we all feel like everyone is always out to get us, makeing it hard to be social. So if you feel like everyone has a reason for wanting to talk to you to harm you, then you might have it. Otherwise, your most likely anti social.