They can check what ever you want them to check. You need to reverse the roles a bit, though, and TELL them what you're going to have checked. Insist on it. If he won't check free test and E2, challenge him to give you a good explanation for why not, and regardless of his rationale there is no good not to test for those things. Seriously -- it's no skin off his back and there's zero danger or risk to you, so WTF?

Please post up with results when you get them. 24 y/o is unusually young to have low test, but I'm hoping you get to the bottom of what's going on soon.