I have an answer.
To recount my experience, after weeks of not having time to address this problem, two Saturdays ago, I decided to tackle it. I had 2-1/2 hours on the phone with "senior" tech support, at the end of which they told me to go to the store. Another 2-1/2 with a "senior" genius resulted in nothing. Another 2 hours at home, nothing. Two more conversations with Apple which said they had escalated my problem to engineering, but I'd most likely have to wait for the next IOS update. (Great Saturday, right?)
Light bulb went off. I went back to Apple Store and asked to try my imessage and facetime login on a new ipad that hadn't been updated to 8.3. Worked perfectly. They exchanged my ipad mini 3 for a new one under warranty. I'm now using a new ipad mini 3 which I WILL NOT update.
The problem is the 8.3 update and using any apple ID that has been created or modified since then. I told Apple "senior" phone support, and they said it was MY OPTION to update the software. Seriously?!? This is my fault?!?