We've always said that even if you're on the CoC you COULD be summooned off with the Arena adventure card. Is this correct?
No, Arena reads:
"You may summon 1 character in this Region to fight." (if you somehow got the Arena card to the Inner Region, then it would work).
Also I assume that if you're on the CoC spells like acquisition, mesmerism etc work as well.
I guess it goes without saying you can use acquisition on the actual crown of command that a character AT the CoC is.
Yes, you can target (assuming there is limitation on the spell) a character at the CoC. However, there is no "actual" Crown to Acquisition, the Crown of Command is just a game mechanic allowing the casting of the Command Spell.