Maximum hand size

I was wrong about this rule. You only have to discard when your hand contains more than 7 cards at the end of your own turn, during the cleanup step after the end step.

Quote :

514. Cleanup Step

514.1. First, if the active player's hand contains more cards than his or her maximum hand size (normally seven), he or she discards enough cards to reduce his or her hand size to that number. This turn-based action doesn't use the stack.

A question about this rule: If a card is discarded from your hand in this way. Let's say you have a card that has some kind of ability where if it's discarded from your hand it's ability activates. Would this be a valid way to activate it?

Yup.