You might be right, but flipping the sign of the y coordinate in equations 1 and 4 puts the ground back where it should be, on the bottom. (unless you are in Australia :P)As far as I can tell, flipping the roots of phi corrects all the problems and implies that the ground is on the top. There is no slipping and the fastest part of the hoop is on the bottom, away from the ground.
I don't think there is any problem with eq4.
By doing that I have produced an applet simulation using your equations that behaves perfectly well with the hoop rolling from left to right in the frame S' with the ground on the bottom and the hoop rotating clockwise in both frames. I am just looking for a friendly webpage that supports java applets to upload to so that you can check it out. The source code for the applet will be available so that people can check there is no "trickery" going on. The applet would not have been possible without your excellent work in deriving the equations ;)