If it is that painful then you probably don't have a decent cup. Generally speaking a referee can tell if you have a cup on especially if you are wearing tight, spandex type wrestling shorts and don't even have to check beyond just seeing you have one in. Cups are like bras for women-once you find one that does the job well and is comfortable to wear then treat it like gold because if you lose it then you might be pretty uncomfortable for some time until you find another good one that fits you well.

Legendary kickboxer Bill "Superfoot" Wallace lost a testicle from a hard kick to the cup in a kickboxing bout. While athletic protectors do offer protection, if you get kicked at the wrong angle and someone's foot drives the edge of the cup into your ball bag, then you could lose a nut just like Bill Wallace did. Ouch!!!

yeah they can be. Depends on how they do them and your cup better be placed right. If you're talking about a full force hit then hell yeah. it probably wont hurt if its a light tap to see if its on.

The rim of the cup still digs into the pubic bone so yeah, even with a cup it still hurts.

yes

But not as painful as the alternative...