Um this is more of a veterinarian question. I am worried about the health and safety of my cats..pls don't take this down

20 POINTS!!
So I just got a new kitten (Lily) but I already have a cat and his name is Sam. Sam is 2 and lily was 5 weeks when we got her. This is her third week with us. Sam has already gotten used to her and her smell. Sam and Lily have interacted through the carrier and have been playing. We put them in a room and they started play fighting. There was no hissing, growling, and their ears were perked up, not flat or backwards. He's been very protective over her and urges us to get her when she is meowing. She has her own nursery and they have played footsies under the door twice already. We thought she was doing okay with Sam so today we decided to let her out in the house with him, with our supervision of course. But she took a shower today since she had an accident on her nursery floor and stepped in it so her paws were full of her feces. She was still a little bit wet when we put them together and while she was showering he kept checking on her since she was not happy about her bath. He sniffed her like crazy when we put them together and then licked her. He also did a biting motion that kind of looked like he was trying to pick her up. I think it scared her and she went under the couch. He followed her and made cooing sounds. Every time she ran somewhere he followed her, even when she went to her litter box. He kept chasing her and idk if that is good or bad. She ran behind the counter where my bf was. Then he said that he "bit her neck and she yowled". we took her away instantly. I looked up what biting on the neck means and it said that that's just how parents pick up their young but idk why she seemed so frightened. We never actually let Lily roam around the house on her own, this was her first time with Sam. I have no idea if this was aggression or not, I need a pet specialist because I don't want Lily getting hurt. She is VERY small.