You can take the idea from your example and add one more pin by putting the pieces closer together and shifting them down so that it becomes possible to pin the last pinning rook:
Then you have four pinned pieces, using a diagonal and a rank/file per king. Intuitively, I don't expect it will be possible to have a longer chain, but I'd love to see a counterexample!
Regarding the second part of the question, yes, it is possible for all the non-king pieces to be pinned. Not a very exciting example, but imagine the two kings and a rook of each color all on a single file, so that the rooks pin each other:
I can't think of a way of doing this with more than two pieces.