Hello Chris,
I remember you posted in the SportscarForum (SCF) on this topic but you did not get a very good reception there (and I did not have time to reply but I really wanted too). I do not own an Aston Martin DB9, however I am a very big enthusiast for Astons.
You might want to ask around a bit more, since there are probably people that know the DB9 much better than I do (and actually own it) but I believe that is the right way to change from the auto to a manual, you should pull backwards on one of the shift paddles. However it does depend on how much RPM you pull the paddle because if you are in a low RPM range in the previous gear (the one you had before you engaged 3rd) you will enter 3rd gear at a low RPM (especially if you go form a lower (1 st,2 nd) gear to third. Can you please describe how it behaves a bit more clearly and describe EXACTLY how you shift before going into manual or third? The touchtronic 2 gearbox in your car should be great though.
The "Sport" button would in brief make the car a bit more aggressive. It would stiffen up the suspension, give you a broader rev and torque band, make the shifts quicker, and more aggressive, and will get the ecu of the car (By ECU I mean the Engine Control Unit, and not the "key") to respond quicker to throttle input- the opening of the throttle valve will be changed slightly for a specific RPM value/throttle input value to allow more oxygen into the engine at times (but not very significantly). As a result of these small changes the car becomes more sporty and aggressive as hopefully you noticed from first hand experience.
I think this is all I have to say about your DB9 right now, maybe I will come up with a few other extra things later because right now I do not really have time to talk for really long... but I hope this clarifies things for you.
You have a great car (despite what some of the members at the SCF said) with (in my opinion) one of the best engines currently available for a grand tourer ( Flat torque curve, low rev, natural aspiration). Enjoy your car.