Yep, Friday Night Fights they're called. They start around 9:00 pm and its mostly Wolves Den runs (synched, aka yellow box on the side). There's also seize players too during that time. When you mentioned olympics though I was thinking something like some players would wear specific glamoured shirts and run around doing interesting tasks involving platforming, enemy killing (like kill more within the alloted time or something) or sprint around eorzea as a naked gatherer class till you make a full circle, or to a certain point, idk. That'd be pretty sweet X)

I wouldn't mind a tower raid event though or the other stuff you mentioned. Incentives to do stuff is always nice, especially in this game :/