what are pros and cons of an air handler located in cooled space?
This is the same arrangement as in all 270 houses in my tract development in Dallas TX. It is the standard arrangement. There is some insulation on the inside of the sheetmetal panels.

EDIT Given the new info that this is a heat pump it would be some benefit to put an insulated ceiling on the closet. Possibly the original heating was a gas fired furnace and the heat pump a retrofit.

EDIT2 If there is a gas fired water heater in this closet,then it is probably necessary to have an opening for combustion air or to expel heat to the attic.

EDIT3 Our closet has an insulated drywall ceiling with one small section of screening to supply combustion air from the attic to the gas fired furnace and the gas fired water heater.