For example, Facebook app and Words With Friends app. If they are closed, will I receive push notifications, if the notifications are turned on in the app and also in the general push settings?
I received a Facebook message about 20 minutes ago, but did not receive any notification on my phone. When I went into the app, it showed I had a message, but it didn't not vibrate or give me an audio alert.
Just don't rely on the Facebook app for notifications.
If you have your e-mail setup to push directly to your phone. Or even if Facebook is setup to send you text messages, those are more 'instant' ways of getting notified of updates. But until the push notifications on the iPhone app are fixed... Don't rely on it.