But you can login to your wireless network and the internet with no problems? And use other internet functions?
Curious since I use a regular iPad and have no isues like this, but don't have a mini.
Have you tried Software Update and App Store Updates to see if there are updates to the software for the mini?
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
Using FaceTime and iMessage behind a firewall
iOS: About Messages
Set up iMessage
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
For non-Apple devices, check out the TextFree app https://itunes.apple.com/us/app/text-free-textfree-sms-real/id399355755?mt=8