If part of their childhood is spent being taught to make silk carpets, I would say, "According to the tradition in this region, young girls are brought up learning how to make silk carpets". This means that they are continuously taught the steps and skills needed as they are growing up.

Yes but I would say it is a little formal and old-fashioned. Here, people are brought up to respect their elders. Men in this family are brought up to join the service. (which means "enter the military")

It sounds natural and correct to me.

It's not correct because it could imply that the reason why they are brought up is to make silk carpets which doesn't seem to be the case. It would be better to say 'young girls learn how to make silk carpets in these regions.'