Here are the results from my appointment at the Apple Store. The Genius quickly decided that the phone was broken and looked up the warranty status. Since this was a replacement for a previously damaged phone, it was out of warranty even though it was under a year old.
The Genius immediately attempted to upgrade me to another iPhone. I said no thanks and today bought a Samsung Galaxy S4 Active. Hopefully it will be more robust than the iPhone 3GS phones I had.