Hey Zeus--,
I can understand how frustrating it can be to have a new device misbehave so soon after receiving it. Notification bar and Control Center issues are typically caused by some software abnormality. Fortunately there are a few relatively simple software troubleshooting steps that can help stabilize your iPhone.
It may help to simply force close all open apps and reset your iPhone.
Force an app to close in iOS
http://support.apple.com/en-us/HT5137
Turn your iOS device off and on (restart) and reset
http://support.apple.com/en-us/HT1430
If the Control Center and Notification bar continue to cease functioning, the next step would be to reinstall iOS on your device by placing it in Recovery Mode, then choosing Update in iTunes.
- Make sure that you have iTunes 12 or later on your computer.
- Put your device in recovery mode.
- When you get the option to restore or update, select Update. This will reinstall iOS without erasing your data.
If your iPhone, iPad, or iPod touch doesn't respond or doesn't turn on
https://support.apple.com/en-us/HT201412
Sincerely