The first thing to do is to remove your battery while your phone is powered on then replace it.
Now that you've done that, see if your phone runs smoother.
You also said that you can't access App World on your phone. Can you offer some more information? Is there no icon anymore, or there is but it doesn't respond, or are you getting an error message?
Please post back with updated information. 
Ok, we're going to uninstall your App World then re-install it. From your homescreen, go to Options>Applications. Find App World, highlight it with your cursor, press the menu key and select Delete. Once it's deleted, from your phone's browser go to www.blackberry.com/appworld and download it once again.
Please post back once it's been installed and we'll see how you're doing at that point.
Re: Updates
OK, on to the next step.
You've already pulled the battery multiple times, so everything SHOULD be running smoothly. If your App World just keeps saying "loading", there is obviously something up with it.
My suggestion is to reload the OS to your phone. If everything else on your phone is working properly, this might seem like overkill but it's what I always suggest after battery pulls and it's helped a few people.
Plug your phone into your PC and open BlackBerry Desktop Software. Once your phone has been recognized, click Device at the top of the software then Restore. Choose the most recent restore point and load it. This will reload your OS including all the important things like App World. I suspect this will solve your problem.
Please post back if you need help with this.
