It's a shame, but the Facebook app is utterly dreadful. Inconsistent, often not working at all. I persisted for so long, hoping that with each update it'd become properly functional, let alone include basic stuff like poking, viewing comments on events (which you still can't...which I find utterly incredible).
I've taken to just using Facebook via the browser. The touch mobile version is great, and far better than the app unfortunately.
In terms of notifications, you have to go into the setting by pressing menu and going in there when on the main page of the app. Again, it's inconsistent and works when it wants.
In terms of posting a video, you can do it by clicking the share button in the gallery, and selecting "Facebook". You'll notice there are 2 Facebook options. One for vids, the other for pictures. Which is which? You have to guess. One option will show vids, the other pictures, so you just have to try one of them and see.
As far as I'm aware though, when you do it it doesn't show a thumbnail, and you can't include a comment/description.