For Facebook, search the market for Facebook for Android. This is the updated version of the stock Facebook. It will save over the Facebook app that is currently installed(it won't erase your data) plus this will alert you when there are new app updates.
For your tasktiller, Froyo has a decent task killer built in. There are always small tasks that happen in the background for different apps. I find that my battery lasts a little longer after I uninstall advanced task killer.
I hope this helped. Have a good day.
DELETE YOUR TASK KILLER WITH 2.2
fb app not working since new update.. whats going on
Please uninstall, then reinstall the Facebook application. This should resolve the issues for you. To uninstall the app, please select Menu-->Settings-->Applications-->Manage Applications-->Facebook-->Uninstall.
Once this is completed, please re-install the application and you should have no problem logging back in.
Please let us know how this works out for you. Have a great weekend!