Can I wire this light fixture set with the plate mounted directly to the stud?
As Jack noted, you really really need a junction box here. As mounted, the wires could rub against the metal plate. A pancake might be overkill, though (and you do need a NM clamp for those). I would recommend a joist junction box instead



It's fan rated, requires no clamps, and you can attach your plate to it easily. Oh, and you don't need to ground a plastic box.


  In the past I've always mounted lights to a box and grounded the box. This fixture wants you to mount their disk and there is no grounding screw.


As long as the fixture itself is grounded (which is has a ground wire in that green/yellow stripe) then the plate will also be grounded through the screws.