I would like to be able to query the status of the "new mail" icon that shows whether there is new comments or favorites activity. Is this possible?
No. The mail icon is for activity associated with a user login - things that have happened since they last visited with an authenticated machine. The API has no authentication - it's site centric, not user centric. Hence, this won't be possible until at least v2.0 (sometime next year).
Can't you get the user timeline of an ID, which will give something very close to the new replies if not the same thing.