No, the Giant Elk is given only one Action (and no Bonus Actions), and thus can only choose to do one of the available actions.
The text for Charge reads (MM, pp325)
If the elk moves at least 20 feet straight towards a target and then hits it with a ram attack...
Your Actions(and specifically those for Attacking) are as follows (MM, pp325)
- Ram
- Hooves
The Charge means you are choosing the Ram Attack.
The next round, if the creature is still prone, you may attack with the Hooves. Otherwise, it's just another Ram attack without the charge (unless you take the OA to move away and run again at them.)
And as Andras says, there ARE beasts like the Elephant (MM, pp322), that have viable Bonus Actions available to them.
Trampling Charge. If the elephant moves at least 20 feet straight toward a creature and then hits it with a gore attack on the same turn, that target must succeed on a DC 12 Strength saving throw or be knocked prone. If the target is prone, the elephant can make one stomp attack against it as a bonus action.
The (Giant) Elk, however, is not one of them.
The PC version's limit does not affect the monster version (ability assumptions like that can fall under DM Fiat, but that's not valuable information if you're looking for an informed ruling).
And the Bugbear text block does not specify a limit - so there is none.
It would trigger once per hit.
However, Bugbears do not have Multiattack - they can only make one attack per round. Thus, "per hit", "per surprised creature", and "per battle" are functionally equivalent - creatures are never Surprised for more than a single round at the start of the fight.
A Bugbear Chieftain or other monster with both Surprise Attack and Multiattack would be able to trigger it as many times as it hit attacks on Surprised creatures.
Best Answer
Yes, the bonus damage is added to both attacks
When an effect can only happen once per turn, its statblock will specify that restriction. For example, the Assassin NPC statblock has the Sneak Attack trait (MM, p. 343; emphasis mine):
(In the first printing of the Monster Manual, this restriction was originally listed as in a parenthetical following the name of the trait - i.e. "Sneak Attack (1/Turn)." - but this information has since been moved into the first sentence of the description instead.)
Since the doppelganger doesn't indicate a maximum number of times it can deal the bonus damage, then it applies to all of its attacks that hit on the first round of combat against a surprised creature.