Trigger wrote: This vampire can lock this card to take an action during any Methuselah's minion phase. Is this action directed towards that Methuselah who's turn it is or only towards his normal predator/prey?
Whose turn it is has no effect on the directedness of the action. The action is directed at whoever it's directed at. If the vampire chooses to do a cardless bleed, it's directed against the acting (Enkil Cog) Methuselah's prey. If the vampire chooses to call a political action, it's undirected. If the vampire chooses to diablerize his grand-predator's vampire, it's directed against his grand-predator. And so on.
Trigger wrote: And what is the timing of during?
This is just the normal sequencing rules, applied to the opportunity to take actions - as opposed to the opportunity to play cards or effects at any other time.
At the start of the minion phase, and whenever an action concludes, the Methuselah whose turn it is gets the opportunity to take the next action. If he or she chooses not to take an action, then the opportunity to take actions passes to other Methuselahs following the normal sequencing rules. If those Methuselahs can take an action (such as with Madness Network, Enkil Cog, the Antou), they can choose to take an action - each one gets the opportunity to do so, following the normal sequencing rules. If any of them choose to take an action, then at the end of the action, the opportunity to take the next action goes back to the Methuselah whose turn it is. If the Methuselah whose turn it is says they're done, and then no-one else takes an action, the minion phase finishes.
This means that for cards facilitating out-of-turn play, the Methuselah whose turn it is can get in the next action if they leave someone untapped / otherwise able to act. For example, if a vampire goes to torpor from an out-of-turn Madness Network Rotschreck rush (a fearsome deck, back in the day), the Methuselah whose turn it is might leave a weenie untapped to rescue the fattie star if it goes to torpor. But if the Methuselah whose turn it is passes on the opportunity to take the next action (with someone still untapped), and then everyone else passes, the minion phase ends - so you might leave vampire untapped like that unnecessarily and not get to act with it, if the Methuselah you're expecting to take an out-of-turn action doesn't.
youd typically want a way to unlock them either in your turn or in someone elses. Various master cards, or freakdrives, or mylan horseed ally, reactions that unlock you etc.
Time to use Enkil cog is when a player is about to move to influence phase from minion phase.
One main reason to use Enkil cog is to bleed during your prey's turn, because then you can pump the +bleed as much as you want and be safe from Archon investigation. (Player cannot play out-of-turn master cards during their own turn, unless the card specifically states so. Archon investigation does not.)
But the action can be any action. You can do undirected actions as well. I once used enkil cog to turn anarch in a tournament, to save myself from hefty anarch revolt damage. Vampire has to be unlocked to perform an action (usually. There is Force of will.).
"Plenty of little men tried to put their swords through my heart. And there's plenty of little skeletons buried in the woods."
- Tormund Giantsbane, Game of Thrones