I'm guessing that you probably have a bad battery.

I have the 1640 mAh battery in my Q.  I wanted to see how long it would
last so I unplugged it at 0545 one morning and used it like I normally
do, checking email every 15 minutes, checking out websites, playing
mp3's, streaming audio, taking a few pictures, some video, etc.., and
used a bluetooth headset when making phone calls.  I left the bluetooth
turned on (on the phone) when I went to bed around midnight.  The next
morning I found that the phone had turned itself off due to low
battery.  I imagine that if I had killed all the running apps and
turned off the bluetooth, it would have still been powered up the
following morning.  Having said all that, I'm getting at least 18 hours
use out of the battery.  The one thing I did when I first got the
extended battery was to put it in the phone and run it dry, then charge
it up for a full 24 hours to give it a good solid charge and run it dry
again.  I do this with all my NiMH, LIon, and Ni-Cad batteries.  People
say that the NiMH and LIon batteries do not develop any memory like the
Ni-Cad's, but I always like to "condition" them to accept good solid
charges.

Sorry for being long winded.  I'd take the battery back to the store
and exchange it for a new one. And to answer your other question, yes,
applications do run in the background.  To turn them off completely, go
to START > SYSTEM TOOLS > TASK MANAGER....   Select which application
you want to turn off and select MENU, then KILL.  Or if you want to
turn them all off, select KILL ALL.

Chris