Is it possible to come back from tj to san Diego only with a copy of my birth certificate and a school ID

You need the original and a picture id. You might have problems going in to Mexico thou. Mexican immigration requires you to have a valid travel document which is a passport. If they stop you and you don't have one they can send you back to the states or they can ask for money.

You need a notarized copy of the birth certificate not just a copy. If you have the notarized copy along with a photo ID, then you are ready to come back across.

Yes

You need the original certificate and you id