This may not help anyone, but I FINALLY got Facetime to work properly on my iPhone 5.
I could receive Facetime calls on my phone from my wife, but when I tried to initiate a Facetime call to her, I would immediately get the "Connection Lost" error.
I tried most of the suggestions in this thread to no avail.
I tried to initiate a FT call while in my car (connected to the radio via Bluetooth), and when the radio displayed the "Calling 123-456-7890", I realized that that was not her iPhone number, but her work number. Had I not been connected to the radio via Bluetooth, I never would have known that the phone was calling her work number (a land line). No wonder it didn't go through!!!!
Her work number was the first number listed under her contact info. I deleted that number which moved her iPhone number to the top. I then added her work number beneath the iPhone number.
I was then able to initiate a Facetime call from my phone to hers.
Again, this may help no else else, but it solved my issue.
Good luck to all.