I just posted this suggestion for someone else ...... This might the next thing to try.
You can try resetting all settings. Settings>General>Reset>Reset All Settings. You will have to enter all of your device settings again.... All of the settings in the settings app will have to be re-entered. It does take some time to re-enter all of the device settings again and some will be re-entered as you go along. It will not all happen at once.
You will not lose app data at all when you reset settings.
If that doesn't work, I would think that the next step would be to restore the iPad iOS software again.
Thank you. I tried the first suggestion but ended up doing a full restore. This has worked in that imessage is now working and the annoying message to connect to iTunes has gone. However, my ipad still can't find the keyboard and I have 14 apps that fail to install. I have tried installing them all separately. They load and say installing and the blue bar gets to 90% and then fail.
Do you restore by erasing device as described below, or did you restore the iOS via iTunes on your computer.
You could try earring the device in Settings>general>reset>erase all content and settings. You will have to start all over again and activate the iPad again. You can restore from either an iCloud backup if you use iCloud - or from the iTunes backup - if you backup with iTunes - and then sync all of your content back to the device after restoring.
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
Troubleshooting FaceTime and iMessage activation
iOS: FaceTime is 'Unable to verify email because it is in use'
Using FaceTime and iMessage behind a firewall
iOS: About Messages
Set up iMessage
iOS 6 and OS X Mountain Lion: Link your phone number and Apple ID for use with FaceTime and iMessage
How to Set Up & Use iMessage on iPhone, iPad, & iPod touch with iOS
Troubleshooting iMessage Issues: Some Useful Tips You Should Try
Setting Up Multiple iOS Devices for iMessage and Facetime
FaceTime and iMessage not accepting Apple ID password
Unable to use FaceTime and iMessage with my apple ID
How to Block Someone on FaceTime
My Facetime Doesn't Ring
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.