No you could not make an exact duplicate of Micky Mouse and get away with renting it out and in fact many costume houses did get in trouble for that. What they do is change it just enough to step into the grey area. It really does not take to much to get there.

That's what I thought you meant - just wanted clarification.
I really don't think that Russell's puppets infringe on the design of the original puppets, no more then Swazzel's take on Audry II infringe on the original puppet.
Can't speak on that: I honestly don't know much about US copyright laws. One thing that I've been wondering though:
I've been sorely tempted to create a set of Audrey shadow puppets. I can't think that it would break copyrights, and unless the licencing for the show specifies a particular design for the puppets, would something like that be ok?
I mean, there's a vast difference between making a Swazzle replica of their Audrey/any other typical Audrey and making an Audrey shadow puppet... It's one of those grey areas, but again I don't know much about these kinds of copyright issues.
Let's not forget the Avenue Q itself is a bit of a copy. It is a parody of Sesame Street. This means that any puppets that would be built for it would need to be that style of puppets and the characters do have specific features that are needed that are outlined in the script.
Agreed. One thing I say a lot on my site is that you can copyright a
character, but you can't copyright a building method. I think it's fine to parody characters or a show, or to use similar building methods to achieve similar character designs; but we can all agree there's a difference between similarity/inspiration and replication.