The 3200 (and 3500) are known to turn on after a power failure, so the first thing I thought of was the power supply - but that was one of the things you swapped out, so I ruled it out. But if the UPS isn't, well, uninterruptible, that would cause all sorts of problems. Not only turning on, but also incomplete recordings.
Maybe it's just a bad connection within that one power receptacle on the APC UPS. If you can confirm that and the APC is under warranty, I'd return it for a replacement. Their equipment is usually pretty good, but I guess every product has its occasional "lemon."