Is this AC only or is a furnace involved? HP?

It is AC only. Not a heat pump.

maybe I am not understanding something correctly.

Wouldn't it make more sense to vent the heat of an AC air handler outside (where you don't care where the heat goes) rather than inside the conditioned space you are spending money to cool?

Since the space is yet to be finished, would it make sense to have the closet insulated, but with an attic vent it it for ventilation?

Again, If I am not understanding something properly, I apologize

Air handlers don't run hot nor do they 'require' some sort of outside air ventilation. They are installed in attics all the time. Important thing is that they are sealed properly. Since the attic is also going to be conditioned space, not as much of a concern for hot attic air getting into the ductwork (though sealing is still a must).