For cards like Sniper, does the active ability cooldown start even if he's not in play?
I notice that when sniper is played initially, the cooldown of its ability starts at 3. If I hold it for later would the cooldown still be at 3?
Yes. Sniper's active ability cooldown starts from the start of the game.
I have tested it myself and the wiki agrees with me.
It is the case for all heroes as long as the hero exists. This means that for heroes like the copies of Meepo, the cooldown will only start after they have been summoned. Ability cooldowns decrease at the end of every round, regardless if the hero is dead, alive, or waiting to be deployed.