If school policy is to not allow parents to wander the halls/accompany their children to class, then the teachers in the hallway were absolutely right to stop the OP and ask what he was doing, because as far as they knew, he wasn’t supposed to be there. Sure the office issued the Visitor badge, but those could be stolen or faked, and he was there under unusual circumstances that went against what all the other teachers knew to be the policy. So good on them for doing their job.
I happen to think the policy makes sense, because having non-employees and non students wandering the halls of a school is a liability. That person could be a non-custodial parent or someone with intent to harm a child or a drug dealer or simply lost or… it’s true that most of the time, a parent is a parent and they only want what’s best for their kids, but unless the school verifies that this is the case, then other possibilities arise, as has been shown through countless past events at other schools across the country.
In this OPs case, what the school should have done is have someone from the office accompany him and his child to the new room, both to be polite and because it would prevent other teachers from having to interrupt whatever they were doing to ask why he was there.
As for punishment: at face value, I think punching someone for pulling your hair is a little over the top, and since the likelihood of getting the whole story from the two girls is rather slim, I think punishing both for breaking a “no fighting” rule is warranted. Especially for 6 year olds, who are still learning proper social interaction and need to learn that responding physically is not appropriate (this is true of the hair puller too…she may have done it because she was jealous of the pullee’s pencil case or something irrationally child-like like that!). There are alternatives to fighting that children need to learn, and this is a good time to teach that. Naturally common sense needs to be applied if the situation escalates to full out assault, but even then, there are appropriate levels of self-defence and inappropriate ones, and even adults are punished if they use inappropriate ones, no matter who “started it”!