Unfortunately, there is no short answer. Each school system has different tests/requirements for services and what qualifies you in one district may or may not qualify you in another. Assuming your child is not in a private/charter school situation, you should be able to get the policy for early entrance from the County/District Office.
For your older child, if you are in the US, there is likely a written policy for AIG/GATE qualifications. I would begin by asking for the tests that are given for qualification. Also, this policy should let you know whether or not outside testing will be accepted and, if allowed, will give some indication of which tests your district will take.
Others can give more specific info about the tests themselves, especially those used to assess learning issues. Best of luck!
