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1. Double check your Facebook notification settings in Settings > Notifications > Facebook and make sure you have sound notifications enabled (although in my example I have it disabled by choice). Also make sure that all alerts have assigned alert tones in Settings > Sounds
2. The App Store only checks for updates every once in a while to preserve battery life. If you open App Store and go to Updates, it will force a check though. If you don't have "use cellular date" enabled in Settings > iTunes & App Stores, updates won't download until you're connected to wifi. This is to preserve battery and data usage.
3. You can always slide up from the bottom of the screen to bring up the control center and then adjust brightness from there. If you have Auto-Brightness enabled in settings your phone will adjust the brightness automatically based on ambient light conditions.
I hope this answers your questions. If you think of anything else let us know. We are all happy to help!