Tranny: You may (hopefully) be experiencing the "marbles in a can" noise reported by many Z06 owners.
Here are some threads:
http://forums.corvetteforum.com/c6-z...ew-08-z06.html
http://forums.corvetteforum.com/c6-z...-in-first.html
http://forums.corvetteforum.com/c6-z...ew-08-z06.html
I haven't done the research, but I am pretty sure the cars share the same, or similar, Tremec tranny.
Some say this is "normal" -- others mention it may be a rear diff fluid change required, and/or a clutch pack change.
I may be totally wrong, but this at least gives you some speaking points to the dealer when you take her in.
Engine shake? It takes awhile for the computer to "learn" your driving style. Have them check it out, but I'd bet the computer will "smooth that out" over time.
EDIT: the sunroof - DEFINITELY pursue that further!!!!
I hope its nothing serious. Good luck, and thanks for reporting the issue.