Do you think he believes in the supernatural?

Probably:

In an interview clip he said that the Warren's were doing what no one else was back then. And nowadays there are other parapsychologists.

And many artists (right-brained) do sense the paranormal more than the usual population.

He is refreshingly open-minded by taking on the Conjuring role. And playing one half of an inter-racial couple in another in Lakeview Terrace. With his liberal views, he is likely to at least consider the possibilities of ghosts one would think.