Does my cub remain on the battlefield or back in my hand?
It remains on the battlefield. Reflector Mage's ability does target ("When Reflector Mage enters the battlefield, return target creature an opponent controls to its owner's hand.[...]"), and Hexproof makes that an invalid target:
CR 702.11b “Hexproof” on a permanent means “This permanent can’t be the target of spells or abilities your opponents control.”
The Reflector Mage, and its triggered ability, are both controlled by your opponent, so the target is illegal. Since the ability has all illegal targets, it is countered upon resolution, and none of its effects happen.
My Bristling Hydra is on the battlefield. I spend energy to make the hydra hexproof and +1/+1. My opponent casts Reflector Mage, targeting the hydra.
This cannot happen as described. If you have activated, and resolved, the hydra's ability, it cannot be chosen as a target for Reflector Mage's ability.
That being said, if Reflector Mage is entering the battlefield in response to your hydra's activated ability (perhaps your opponent cast Collected Company in response, and found a Reflector Mage), then yes, it can be chosen as a target. Unless you choose to respond to the targeting by activating the ability a second time, then the hydra will return to your hand.