No heat source will go through wood. What you have to do is install the flex INside the cage, then put vinyl flooring over it and silicone seal the edges. The cord for the heat source (and the probe for the t-stat) can go through a hole cut in the floor of the cage, which will be covered over by the vinyl.

To treat the wood, use about 3-4 coats of Polyurathane (one coat of polyShades if you want color). Make sure to let it cure in the sun for a few days before you start using it, though. Not sure about melamine paint, but you would probably have to let it cure outside longer.