Child 13 years. bottom left tooth with small brown specks possibly cavity. would general pain in surrounding gum be normal with cavity?

Child cavity: It's hard to say if your child has a cavity based on the description of having brown spots without a first doing proper dental exam. It is not ever normal to have pain in or around the gums or teeth. If the tooth has a deep cavity, the infection can spread to the tooth pulp/root and supporting bone, resulting in an abscess with tenderness/swelling. You should have your child seen by a dentist asap.

Cavity : Cavities usually do not cause gum pain unless they are already into the nerve which would be a very bad cavity. I would try to see a dentist to get x-rays and a visual exam as soon as you can.

Needs examination: Firstly, we cannot tell, over the internet if this is a primary tooth or an adult tooth. If a primary tooth, the secondary tooth may be trying to erupt, causing the gum pain... This is only a guess. It is really best that the child sees his/her own dentist.