Yeah hachimaki is basically a headband that Japanese people wear during various sports events at schools, some marshal arts classes or local sports days. At some schools, Kids wear them during P.E. classes (sometimes red ones for girls, white ones for boys). I think the original purpose of hachimaki is to stop sweat from coming onto your face, but now it's just a routine thing with a connotation that you're ready to fight. When we see a picture Dana posted above, we know that he's ready to fight---maybe not literally fighting. It could be studying hard for an exam or starting to do a dangerous adventure.
My friends and I used to joke at times that somehow Gonzo's and Nalby's headbands look like hachimaki. Don't know why, it could be the length or in Gonzo's case he looks he's always ready to fight!
As for Kei's "hachimaki," I think it looks fine and as chewy said it's better than the backside cap that fell in Houston. But in Japan, UNIQLO still represents comfy but cheap clothes so I'm not sure if it is cool.