you could just cut an inch or less off the bottom of the door if you didn't want to look a grill messing up the appearance of that beautiful wood door. A one inch gap at the bottom is about the same as a 4x8 hole in the door. Looks like you might already have somewhat of a gap at the bottom now which is common so you just need to make it a little bigger. I would install your other two parts of the plan, and see if you need more of a gap to keep the closet cool. It is generally against code to run an exhaust fan into the attic space. You should connect it either outside, a return duct or another exterior exhaust duct. If you tap into a bathroom exhaust you should install back flow preventer dampers for obvious reasons. You should worry about lint from a laundry exhaust duct. The easiest is just take it directly outside, The most energy efficient is to connect it to a return air duct.