One of the most memorable lines of my childhood is from the leadup to the Ahavas Yisrael song in the Marvelous Middos Machine (tape 3). “Look over there. There’s that boy with the funny-looking knitted Yarmulke. And he’s carrying a Gemara. Why would he need a Gemara?” It’s sad that some things haven’t changed.
I have not yet found a compelling Halachic reason to wear a hat. I do know that R’ Schachter says it’s completely unnecessary nowadays (he still wears one though).