[rodmillard]

I don't have the book in front of me, so I can't check the item description, but I can't think of any reason why not. The FAQ states that the player whose turn it is may choose when allied wizards cast their spells - I imagine the same applied to a character with the Wizarding Hat (or they cast their hat spells when they would use their incantations).

As for determining the lore, personally I would shuffle together the magic deck and randomly draw one spell; you get that and one other from the same lore.