It depends who's massaging you - I imagine it'd be easier without clothes, but I doubt anybody's going to force you to strip...
During certain types of massage you can such as Thai massage, Shiatsu or chair massage.
DICK?
Generally you are nude, but you can wear underwear, especially for women who are menstruating.
You wrap yourself in a towel.
Yes, you are allowed to wear whatever makes you the most comfortable. There are massage styles where the recipient remains fully clothed the entire time.
You're allowed to wear absolutely anything. :)
When you get a massage of any kind, it is not necessary to be disrobed of all your clothing. Not even a French massage. Some modalities are performed more easily if the cloths are removed, but this is not a requirement and no professional massage therapist will ever force their client to do something they are not comfortable doing, even removing clothing.
You can wear whatever is comfortable for you.
Generally you are nude, but you can wear underwear, especially for women who are menstruating.
generally they were allowed to wear their own clothes
Sure. There is no problem leaving your underwear during a massage.
Unless you are having surgery on your head, there is no reason why you can't wear a wig