Other mistake: They act as if Carson was part of the dojo with the whole crew, but Kim was a Black Dragon when the show started.
Other mistake: They say that when Jack is back, they will be in college, but they were in Year 8 and all 14, so they would be, at most, in Year 12.
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.