If I use 'Staff of the Ogre Magi' and 'chromatic cloak' does the chromatic cloak enhance the damge that my hero deals with the staff?

As long as your not swinging the weapon itself, but using the 2, exhaust, exhaust hero ability, then yes it does enhances the weapon damage. However im not 100% sure.

The 'power' on the staff wouldn't trigger cloak, 'cuz claok only triggers off of 'abilities' which are the ability cards (ex: fireblast, quick strike, etc.)

*Please note, im not a judge. I could be wrong!

The effect text on the staff is the staff's "power"

Abilities are types of cards, not card effects