How should I extend a concrete patio under an existing deck?
Sounds to me that you have covered all the bases. Maybe you should pay attention to few details:

Connecting old and new concrete; you have mentioned that you plan to do it, make sure that they are well stuck inside of old concrete (you drill slightly smaller hole than bar and hit the bar it with hammer when placed inside a hole. Also, you can chip old concrete (inch or two) so it will connect better with new one.
When it comes to posts I haven't seen the picture so it's hard to tell, but if you want to separate it from slab, you can do it by placing thin Styrofoam around posts before pouring concrete. Later on you can leave it or take it out and put some commercial products (make sure that those are for outdoor usage).
When it comes to changing the slope I really don’t see why would you do it? Accept if you don't like it. Anyway, if it's not causing any problems my advice is not play with it....I’m not sure it can be done properly.