For the Mesoamerican ball game, am I understanding the signs correctly? That the losers might be put to death, or is it just metaphorical? Also, did players really wear stone belts?
While that may have happened in some cases, the ballgame was also played for recreation. Scholars have found that one city in Mesoamerica had at least eleven ball courts and the ballgame was very popular throughout the region for centuries!
The stone belts are for ceremonial purposes. It's not likely players would have worn them.
What kind of ball did they use to play the game?
The balls were made from a natural rubber! They were extremely heavy and players had to wear a lot of padding to protect themselves. This stone piece represents a real piece of athletic gear that would have protected the hip and been made of leather or plant materials.