I just called Delta and United (for what it's worth), and confirmed with my brother who works for one of those two major carriers. Neither airline requires *any* kind of documentation for minors traveling with their parents (meaning: they don't need their birth cert or proof of age). It's a different story if it is a minor traveling WITHOUT Mom & Dad.
I've NEVER been asked for proof of my DD's (5) age or that she's "mine" (yes that part IS kind of scary but if you look at her and then me, I can't deny her!). She was asked when she was just about to turn two how old she was, but that's it.
That being said, I only called two airlines and there are several out there. I don't know what each individual airline's policy is on this, so I stress again that you should call your own airline to verify their policy.
And I personally would never bring any EXTREMELY important documentation with me when I travel - birth cert, Social Security card, marriage license, etc. If that information ever fell into the wrong hands....... It's the same reason I don't bring a purse with me and only carry my ID and 1 credit card in a slim wallet that fits in my pocket.
Can you tell that this topic hit home????

I'm down off my soapbox now..... Sorry!
Lynne