I have discovered that I have a leak. I can hold my nose, close my mouth, and blow and get air to come out of my ear !!! What the hell?? Is my eardrum blown? I just went to the ENT two days ago. WTF? Is this normal?
It is indicative of a perforated eardrum. Maybe a small hole, maybe a big one. May even be a flap that’s not easily seen with a basic ear exam. Ask your ENT about it.
… you’re not supposed to be able to do that?
Come to think of it, I can only do it out one side.
Isn’t “blow it out your ear” a euphemism for the more vulgar expression?
Are you, by any chance, a cartoon character?
I sometimes can do that, but it takes a lot of pressure and it comes out very very slowly. The only reason I tend to get into this situation is that I’m trying to blow my nose and one side is absolutely not budging.
Also, when I was younger I felt some air coming out of my tear ducts in the same situation. Not sure about the mechanics on that one.
Cecil discusses Eustachian tubes: What’s going on when your ears pop? - The Straight Dope
I’m wondering if going underwater might be a bad idea. If air leaks out, can water leak in?
Yes. AND cause significant pain and damage, too.
To the OP: Get in touch with the ENT who just saw you. It’s not normal, it means a perforated eardrum, and that’s all you really need to know.
Thanks, Doc.
I occasionally blow air out of the tear ducts as well, if I blow hard enough 
It feels a bit… weird…, but as far as I understand, perfectly normal given the internal mechanics of the human head - however it is not advisable to do so.
Not so fast CC! There’s a potential novelty gold mine here. For instance, can you stick a kazoo in your ear and play it? Ok, how about a trumpet?
I perfed an eardrum years ago when I flew with a cold. Pain in both ears during the landing went up to screaming pain on one side and the… stopped on that side. Whistling with nose blowing after, so I backed off on that. Took a while to stop, but it did, no comments from my doctors when they look in the ears (years later).
There is a duct that drains the tears into the nose (which is why we get sniffly when we cry or almost cry). For most, it runs one way, but not for all.
Another tube supposed to run one way, but occasionally can run air backward, is the parotid duct (Stenson’s duct). Some people will reflux air into their parotid gland (in the cheek) when they blow a horn or balloon.