J:
I've also wondered about these phrases, though never to the extent you have, and I have always felt there was something I'm missing.  You are alos missing whatever that is.
But your post made me think some more about it, and I wonder if what we have here is a case of ellipsism.
Take, for example, the sentence, "I've got to get a haircut."
Is it posible that in its entirety the sentence reads something like this:  "I've gotten the obligation to get a haircut"?
Similarly, you can make more sense out of "I've got a headache" by thinking of it as the more grammatically correct: "I've gotten a headache."
It's so nice to see a post qbout words!
TEd