Using FaceTime http://support.apple.com/kb/ht4319
Troubleshooting FaceTime http://support.apple.com/kb/TS3367
The Complete Guide to FaceTime + iMessage: Setup, Use, and Troubleshooting
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To send messages to non-Apple devices, check out the TextFree app https://itunes.apple.com/us/app/text-free-textfree-sms-real/id399355755?mt=8
How to Send SMS from iPad
You can check the status of the FaceTime/iMessage servers at this link.
I just wanted to make sure that you did not have system sounds muted. Indulge me here and double check the setting.
Settings>Notifications>FaceTime>On. Select the number of recent items that you want to show. Then go to either Banners or Alerts and select one of them. Then go down to Ringtone and select the sound of your choice in the next window. Select "On "if you want the alert to show in the Lock Screen.
You could try turning FaceTime off and sign out of your ID and start all over again.
Have you tried closing FaceTime in the recents tray, reboot the iPad and see of it works after you do that? Of you want to try this, Her is how to do it.
Tap the home button once. Then tap the home button twice and the recents tray will appear at the bottom of the screen. Tap and hold down on any app icon until it begins to wiggle. Tap the minus sign in the upper left corner of the FaceTime icon. Tap the home button twice.
Reboot the iPad by holding down on the sleep and home buttons at the same time for about 10-15 seconds until the Apple Logo appears - ignore the red slider if it appears on the screen - let go of the buttons. Let the iPad start up.