This tablet can run Facebook if you install it from the 1mobile market found here http://www.1mobile.com/app/market/. Unfortunately, the tablet kept crashing for me and downloads from the markets seemed to stall or not work most of the time.
My 10" Coby MID1042-8 does run facebook. However it does not run all the games, and can take a long time to download pictures or will not load them at all. Also it looses connection with my wireless router pretty often. It seems to work best for light web-browsing, email, and reading apps.
Yes it will....Download Amazon App to your tablet and you can get pandora, FB, Twitter, angry birds, etc. The App store it comes with is not really good...but it takes 2 seconds to switch to Amazon Apps.