No one runs it, it's running itself. SHIELD is all but babysitting it, really, and keeping an eye on who it pulls through. It's an artifact of some kind, but my only interactions with it have been when I was pulled through. It's apparently whimsical - it sent us Waldo from Where's Waldo the other day. Just what we need, a whimsical artifact.
I haven't seen hide nor hair of members of either Court, nor any other creatures from NeverNever, or really anything that relates to home except us. I doubt even the Queens are this creative.