Hat
The person who answered before said 'baseball cap' but baseball is not a British game. It is a Cricket Cap. Cricket is a game played in the UK.
You can, but most players don't. If the helmet is too big, wearing your cap under it is a good way to make it fit better though.
Turbans
New Era
History has not recorded that information although it is known that a team in the 1860s called the Brooklyn Excelsiors wore caps that looked somewhat like today's caps in that they had a rounded top. There were no rules back then that stated players had to wear a cap so most did not wear one although some players would wear a straw hat with a large brim to shade their face from the sun.
The bill.
to keep the the sun out of the players face so that they can focus
The player, himself, decides what hat will be worn.
Cap Anson Mordecai Brown
Button-Up Jersey, Baseball Pants, Long Socks, Cleats, Baseball Cap, Etc.
to be noticed as the one in charge.
Even though the movie was set in Boston, Jack Nicholson didn't want to wear the cap for the Boston Red Sox, so he wore his own NY Yankees cap.