I was working off comic pics from GoogleImages since the Dooms on the show is too shadowy to use as a reference, and while I was surprised as you were, he did indeed have shoes in the image I found! I was like, "Oookay, let's just roll with it..."

On SV he definitely keeps Davis' pants, though in "Legion" they got ripped to the point of very near indecency once he changed back in the Fortress. :D