You are correct to be concerned about sanding plywood. The exterior layer is often quite thin, however it shouldn't be that thin on an exterior grade plywood. Get in touch with your FIL and ask if he remembers what kind of plywood he used. I'd guess that it's an AAX - "A" is the surface grade and means that it's got a nice, sanded finish and all voids are filled - on the right-hand gate, just above the center lock, there's a knot and an oval that doesn't match the grain - that's a filler piece in a void. The second "A" indicates that the other side is also an "A" grade finish, and the "X" is an exterior grade indicating that they used waterproof glues to laminate the plys. It may have also been pressure treated to protect the wood itself from water.
Honestly, thought, there's got to be something other than just plywood going on. Each main panel has very distinct vertical lines, and thought the grain mostly matches, there are changes. Is it a bead board with those thin vertical lines cut into the surface?
To prep the plywood
Wash off the chalk - paint over the chalk will just fall off when the chalk gives up its grip, then you'll have more "permanent" drawings in those patterns. :)
- Just some water should be sufficient to get the chalk off. You could just rub, but you might end up with splinters.
Remove all the mounting hardware
- It will be easier to paint if you're not trying to mask/paint around the hardware
- Water will get behind the hardware and that's where it will rot out first.
Give the whole thing a light sanding.
- Your goal is to remove the surface dirt and to lightly smooth the surface, removing any small slivers that may have been raised by the rain, not to get to shiny, new looking "brown" wood instead of the grey weathered surface.
- You'll want a finer grit of sandpaper, in the 100-150 range. This will help in removing less wood.
- Use a sanding block so you've got a smooth surface
- Don't sand any one area too much or you'll go through the surface layer
- You're probably better off with too little sanding than too much, though this isn't a piece of indoor furniture, so minor flaws are probably acceptable.
Give the solid wood a nice sanding to smooth its surface and remove any gunk
- You can be more aggressive in sanding here if you want, but, you're planning on covering it in paint, so it doesn't need to be perfect.
At this point you should be ready to apply your finish according to the instructions on the tin. Try to get the paint into the mounting holes to cover the interior surfaces, but you don't want it sealing over the openings, though that's a fairly easy fix. Once the paint is dry, rehang the gate, enjoy, and start making plans to paint again in a few years.