Is my cat being mean?

Question

I’m a new cat owner and have only owned dogs so cat behavior is kind of a mystery to me. I recently rescued a 5 year old tuxedo cat from a shelter. They said he was a stray but he’s a very intelligent stray when it comes to the house and due to this they didn’t really have a background on him. He was super friendly when we first got him sought out rubs and purred. On the same day I got him my sister also rescued a 1 year old calico female who was very skittish. He was ready to bust out the door the first day. Then we would let her out at night and him during the day until one day they accidentally met because he would constantly wait outside the door my sister cat was behind and she would meow to get out. After meeting her his behavior changed. He doesn’t seek out rubs and will often try to catch your hand and try to bite you (granted not hard enough to break the skin). Every time he sees her he stalks, swats,jumps at her and then chases her when she runs away. She will hiss, growl and make all kinds of noises at home but he relentlessly goes after her and then will run away. But they will also bump noses and she rubs up against him and they will be fine and then he goes and does it again. After they fight we separate them but then she try’s to come into my room. Neither of them have airplane ears or hunched backs. I’m not totally accustomed to cat behavior. He also flicks his tail all the time with or without contact. His pupils have contracted to little slits (which I think means he’s comfortable) help me!