Some do, but most do not. While a cup is certainly a prudent precaution for professional players, the chances of getting kicked in the gonads are rare, and even so, would carry a severe penalty. Many soccer players do not wear a cup for the same reason that some police officers choose not to wear a bulletproof vest - comfort, flexibility, and mobility are all restricted.
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I think that most Basketball players just wear a regular kind of a jockstrap without a cup. However, some high school basketball players wear a cup because they are concerned about getting hit in their sensitive area. When I played basketball in my teens, a couple of the guys on our team awlays wore a cup.
They should, and the catcher normally does. It's usually a safety rule for catchers in High School and College Sports.
Athletic supporters (also known as cups informally) are not required as part of an MLB uniform. However, it is suggested to wear one to avoid serious injury.