NO YOU WOULD NOT BE MADE FUN OF AT RODEOS!
TONS of people who don't use helmets use them for there runs. It's very likely you will see lost of others wearing helmets.
Some places (really depends where you are) make you if your under 18. Not sure about the rodeos, but at some of the jackpots around here they do.
I use one if I'm with certain people (out of respect), at shows/riding centers that require them, and on a horse I'm unsure of or don't trust as much as some. Otherwise I don't use them if there not required.
You can get western style ones.