You can pay the cost of "discard your hand" even if your hand contains zero cards.

If Connecting the Dots leaves the battlefield before you activate its last ability, any cards exiled by its triggered ability remain exiled face down for the rest of the game. (You still can't look at them.) If you somehow return the same Connecting the Dots to the battlefield, it will be a different object with no connection to those face-down cards.

Each Connecting the Dots you control has its own set of face-down exiled cards. Connecting the Dots's last ability puts only those cards into your hand, not those of any other Connecting the Dots.