Definitely yes! This is a great amenity for residential real estate you'd want to show off.
Not necessary, but up to you.
If there's a lot of stuff there already, then make things easy for you and just leave the door closed. You can add a Mattertag Post if there is something special inside to call out.
If the closet is empty, then scanning inside of it can get you a slightly better Floor Plan and Dollhouse View. However you might want to hide that scan position so people don't walk into the closet.