Does League of Legends have a pause game option?

It does. If you set up a custom game you can select an option called "Tournament Draft". The only difference between "Tournament Draft" and "Draft" is that you can pause the game for up to 60 minutes.

Nah this usually happens when servers are lagging for example once everyone in my match got DC because of the server and when you reconnected it would say "(summoner name) unpaused the game"

Well, actually, if EVERYONE in the match types /pause, it WILL send a pause request to the server. I've had friends who have successfully done this.