I have one of those too, two actually ( it split shortly after bringing it home ). I have had it about 2 weeks and haven't seen any of the polyps like that.. only small brown "holes" or "dots" and the occasional "small spike coming from them. It came from a tank with MH to my PC tank, so I am...
yes you can move them to another rock. when you remove them if you leave any of the tissue from the foot there it is likely they will regrow given time.
i periodically cut shrooms in half ( sometimes quaters ) and let the pieces grow out into new full shrooms.
i agree with doing whatever...
I have cut a few of my shrooms. From what I understand it isn't necessary to have a piece of the mouth but it regenerates faster if you do. I have heard of people literally putting them into a blender with great results.
I cut mine in "pie" shaped pieces as mentioned and they seem to do great...