The lips must touch one another so that they can open and close like a switch. They must be "free" to contract and expand as the range being played requires.
The rim I personally feel is like a good cognac or whiskey - an acquired taste. I firmly believe that the cup volume of the mouthpiece is part of the interface to the standing wave of the horn. For those that prefer a "softer" coupling (with a change in the overtone structure toward less brilliant), it is my opinion that a larger cup volume aids this. For players that want a very direct coupling to the standing wave and perhaps a tilt of the overtones to more brilliant, a cup with less volume could perhaps aid this. This preference only applies to reasonably stong, well practiced players. If the chops are weaker, anything is a crap shoot!