is there a halach or Shilchan orach that forbids giving hair cuts during the 3 weeks?
The Rema 551-3 writes that we do not give haircuts during the three weeks. This applies even to adults giving a haircut to a child see M:B 551-81, that there are two reason why an adult may not give a haircut to a child, either because of chinuch, or in order that the adult should see that the child doesn’t have a haircut because of the aveilus. The difference between the two reasons is with regard to a child under the age of chinuch ( 6-7), According to the reason of chinuch, this child is below the age of chinuch and it would be permitted. However if the reason is so the parent will see it, then it would apply even to children younger than chinuch age.