To seal water spotting or other stains on drywall, use a pigmented shellac primer such as BINS Bullseye. One or two coats will completely seal the stains, then repaint with your regular paint. When using shellac primer, be sure to have denatured alcohol available for cleaning your stuff.
Plywood is a lamination of several layers of wood set with the grain rotating with each layer. This actually helps resist warping, but does not eliminate it (as you can see). This happens because of some flexibility in the glue that allows a gradual creep.
The usual solution is as you suspect - laminate the plywood to something else. This could be a metal bar, either aluminum or steel. Perhaps the most effective would be angle stock (metal forming an "L" in cross section). A flat edge of the angle is screwed to the back. The perpendicular edge then acts as a brace.
If you do this, you need to first flatten the piece as much as possible. I would recommend placing it face down on a flat surface, possibly with a sheet below it to protect the art surface. Then weigh down the corners with very heavy weights, such as full paint cans or barbell discs. Lay an angled bar across the full width near the top and another across the bottom. Place screws every 6 inches. The screw length should be long enough to go through the angle stock and 3/4 of the way through the plywood.
You can then rehang the piece using a wire threaded through holes at the ends of the angle stock. If you really need to use velcro, you could attach wooden filler strips to the upright edge of the angle stock, and attach the velcro to the back of these strips.
An alternative bracing can .be done with strips of plywood, al teast 3 inches wide and 3/4 inches thick. These can be glued and screwed to the flattened panel instead of the angle stock. If you use a strong wood glue, such as Titebond Ultimate, this will help prevent creep, as will the screws.
The advantage the angle stock method is that it is reversible (no glue).
Best Answer
I'd use a few pieces of wood up in the ceiling, to have something for screwing the vent to. The plaster is getting a little worn-out there. I wouldn't trust anchors or make anymore holes for them, nor use toggle bolts; that plaster is about to go and you're about to run out of room (hide-able hole). -I just really don't like wall hangers and I wouldn't want to be pounding some in at that location. It's also not going to be the easiest thing to line up all 3 holes' anchors.
You may get lucky and rotate it just right to find something to bite (don't rip the duct) or just get some shims up there and be done with it. If you do rotate it and find undisturbed plaster to screw to, use drywall screws and not the silver ones you probably have from the vent (larger thread; better grab -do not over tighten them).
Judging by the number of holes, this is the fourth time it's come down; shim it.