Do I need jack studs for a [remodel] closet header that is perpendicular to ceiling joists?
I'd build a stub wall using construction screws (for better pullout strength), and I'd double the bottom plate. I'd then screw (not nail) to the joists through the top plate, and to the king studs through the ends. You wouldn't need to remove drywall on the ceiling if you use at least 3" screws. 

This will carry drywall and trim just fine and will be laterally rigid enough for your purposes. Bifold doors don't normally hang on the header these days. 

Note: If the ceiling drywall is mounted to something other than the ceiling joists, such as to sheet metal angles attached to the wall plates, this is to deal with seasonal truss lift. In that case you shouldn't hang your header from the ceiling joists as it'll possibly move on a seasonal basis, tearing up the drywall with it. You'd want to rest it on the wall using a conventional trimmer setup, and it'll need to be robust enough to span the opening without sagging.