no you dont have to but if youre limited by hair styles and you dont want to dye/etc your hair but you also want your hair to match the costume then it's a good idea.
re: allergies, different wigs are made out of different kinds of plastic, and id recommend researching plastic types if youve had an allergic reaction to plastics before.
but no, you dont have to do/wear/buy anything.
the point of cosplay is to have fun, and you have to make the decision of what you would feel comfortable wearing, how much you want to buy for it, how much time you want to spend making stuff for it, and how accurate you want it to
be/what variations you want to make.
its a compromise to think about, but if its more stressful/effort than fun then dont feel obliged to do everything haha.