From the way I understand it, an Activation lasts from the time you Activate the creature until the time you finish acting with the creature.
Which is the exact same thing as the creature's Action Phase. With that understanding, let's address these questions:
So when you double move as the beastmaster, your turn immediately ends after the second move. Can you double move and use the cowl?
I don't believe so, though the documentation doesn't go into a lot of detail on when exactly your Activation/Action Phase ends. Page 9 of the Arena Rules has this to say: "When the creature’s actions are complete, resolve any effects that occur at the end of its Action Phase...After resolving any of these effects, the creature’s Action Phase is over" This is mostly referring to removing conditions like Stun and Daze, but indicates that your Activation/Action Phase does continue for a little while after you're done taking actions. Thus, I would think that you can still use the Packleader's Cowl after finishing your actions, and would go by the general principal that your Activation/Action Phase ends when you say it does; when you say you're done using the creature.
Begin Creature Action Phase
Quickcast Action
Begin Activation
Activating (uninterruptible)
Choose creature
Flip Action Marker
Remove Guard token (if any)
Action(s)
End Activation
Quickcast Action
End Creature Action Phase
Not quite. It should be a chance to Quickcast followed by Begin Activation/Action Phase. And it should end as End Activation/Action Phase followed by another chance to Quickcast. Quickcast happens before/after the Action Phase, not during.
So over simplification in Arena: Activate -> pause for enemy reveal option -> Mages Activation / Action phase ( i.e. turn) -> End Turn / Activation / Action Phase
Which would mean that Conjurations like ToL happen after a creature has been selected for activation but before it takes its action ya?
Conjurations like Temple of Light happen before/after a creature's Activation/Action Phase, just like Quickcast Actions, so if you want to use one beforehand, you'd use it before Activating the creature.
I really really think people are over thinking this and complicating matters much more than they need be.
A creature's Activation is just synonymous with their Action Phase. Everybody seemed to understand how Action Phases worked before this thread, and they have not changed. This is just another word to reference them, which was only used due to the differences between Academy and Arena. Nothing has actually changed.