Factual error: It says that Jerry is running a judo dojo, but when Jack walks in Jerry is teaching his students to punch - judo doesn't use punches.
Continuity mistake: The lockers in the Bobby Wasabi dojo change from blue to green in the same episode.
Answer: Because he was friends with Kim but he went to the same dojo as the other kids that's how he knows both Kim and the other kids.
No, it never said that. Maybe I'm mistaken, but she wasn't cool with anyone from Bobby Wasabi at the alleged time. So if she did know Carson it wasn't like they were friends, more like enemies if they did know each other.