New answers tagged finish
You bought a pet flap made for a thinner door. There are pet flaps that come made for thicker doors that have an adjustment range that covers the thickness of door that you have. Some indeed will also come with a tunnel piece which gets discarded when installing on a thinner door. I would recommend returning the pet flap for one designed for the thicker ...
I'd try white adhesive tape. Duct tape or electrical tape. If that flakes out after a few weeks or months, then try something stiffer like the Formica idea.
You could try iron-on white melamine banding, usually used to finish the edges of particle board for cabinets or furniture. The banding usually comes in ¾" or ⅝" wide sizes, and it may be a challenge to find the banding in the width you need, but I know I've seen it in a few places.
I think if you use a couple coats of good primer and a finish coat of enamel paint, you should be fine. Just be sure to work it into the wood well. Two thin coats are always better than one thick coat. The cat door is gonna protect the cut edges quite a bit as well. Put a small bead of silicone caulk on the inside of the cat door bezel before you install it ...
Pallet wood probably has some variance in thickness. Also there has to be some slight warping. I really really heavy coat of polyurethane could even out the walking area and fill in the gaps. This is provided the subfloor could stop the liquid flow from seeping to whatever is below it.
Table epoxy comes to mind, but it certainly wouldn't be a cheap solution. Like Michael Karas said, you have to expect a rustic look with pallet wood.
If this is salvaged wood from pallets, you can sand it all down and groove out all the cracks. Then, you can cut little "Dutchmen" to fit those larger cracks, glue them in, and plane that level with the floor. If the larger cracks are sealed, you could then coat the floor with a polyurethane or something similar.
If you built a floor out of pallet boards I would expect that the floor would be uneven, have cracks between boards and rough surfaces. That style is pretty much what you get with this type of material!! There is not much that you could do with it except to rip it out and go with something more conventional. You may be able to cover it with a floor leveling ...
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