So. Much. Ninjaing.
I honestly didn't know it was a sign of respect. I've been to plenty of houses without taking my hat off and no one commented, or even looked at me funny (and it's not just other kids I'm talking about). The only thing I can think of is because I live in the South, and people around here like hats. Probably not it though. Anyway, all of those other things seem natural to me - society never trained me to take my hat off, therefore I won't (or don't want to) do it. Same deal with gum (I was under the impression that teachers/schools don't like gum because most people just stick it under the desk, which I consider far more rude than anything you have described).
For at least 12 years I didn't make eye contact, I sort of just looked at people's mouths. Nothing to do with hats. Guess I'm just an exception to the rule? Still, it's not hard to push the brim of your hat up when talking to someone.
Outside I can wear my hat. Technically, I'm not supposed to do it on school property, but no one actually gives a damn about it outside. I've even talked to the principal while wearing my hat once. I think. Still, at most three teachers/admins would actually say "Take off your hat" if I was outside.