The inside of the coop has exposed OSB ( or plywood? ) and I was wanting to paint the floors and walls in hopes to make it last longer. Though, I don’t know if it matters much, but what color of waterproofing paint would work the best or does it matter? White, grey, black or brown? Some other color?

Then you can white wash it with lime as needed. Another option I've seen discussed here is waterproof rubber coatings, like Black Jack #57.