The Pi is powered through a micro USB connection; I understand you
>need an externally powered USB hub to connect really interesting stuff
>to the Pi anyway. Can you use the same powered hub to power the Pi
>through the micro USB connection and to connect other USB devices to
>the Pi at the same time?
If the hub can put out 1A, then yes. Not all of them can.
I've seen photos of Pi projects running off AA batteries. Is it
>possible/easy to power the Pi from a combination of a powered hub &
>rechargeable batteries so that (like a laptop) it will run for a while
>if the mains power is disconnected?
If you have a battery pack that both charges off and dispenses USB, yes. I believe that one of the denizens of this group is trying to sell such.