You're right in that the New York chapter is meant to be a metaplot that twists between the chapters. You should definitely play it out as much as you can, letting your players have lots of opportunities to interact with people at the Caduceus offices.
There should be plenty of events to make the heroes suspicious about Caduceus. The campaign starts out with the heroes discovering that they've been lied to about the organisation's true motivations. From there, they are sent out to use what is effectively a nuke on a civilian population in North Borneo and to neutralise a largely peaceful cult in Oklahoma for what appear to be ideological reasons. The main hook, however, is the Mafia subplot, where the heroes are pushed into investigating their employers by outside forces. The implication that Caduceus may be involved in smuggling heroin should stir up some interest.
With respect to point 3, there are some guidelines in the text about when the memos about Mu should come into play. If the heroes are poking around the offices early in the game, these memos simply won't exist yet. You are in complete control over when these clues turn up. Don't feel pressured to reveal them early if you don't want to.
The heroes have a chance to encounter Rose Meadham in the Iceland chapter, although not in person. She is speaking to the other serpent people in the control room, using a remote video connection. There are some guidelines there for how she might interact with the heroes.
Hope this helps!