Can you paint water resistant MDF

Normally you can use latex over oil based products (never the other way around) if they are fully cured and dried. If the putty you are referring to is glazing compound, then you may have to wait a bit longer for it to cure. Glazing is very slow to cure and take paint. If you can still very easily dent it with your fingertip, wait, the solvents have not yet evaporated. You certainly can test a small area and see how it goes. However, don't be surprised if you get some peeling in a few months. Oil based paints are hard to find now, but would work better. I don't know what product you used, but there may be some recommendations on the can about painting.

MDF is actually a pretty durable material, and these panels should certainly hold up to occasional hand touch. However, if furniture or other hard objects get banged up against the panel it would be possible to damage any crisp edges, and I don't think there is any finish you can apply to prevent that type of damage from occurring. The finish will make it easier to clean and act as a barrier for moisture.

I think that spraying the finish would definitely be the way to go no matter if you go with clear coat, paint or anything else, since brush marks would detract from the pattern. Considering the cost of the panels, that you probably would prefer to finish them in place after they are installed, and how difficult it would be to perform a 'redo' on a panel with texture, you might want to consider hiring out the finishing. At least experiment on some cheap panels of normal MDF before attempting the real thing.