SITUATION UPDATE! IT'S FIXED!
I have no idea what's going on here, but the original picture & sound breakup originally reported has gone. Everything seems to be working properly (knock on wood). Here's what happened.
For some reason, maybe a glitch in the firmware, the Hopper 3 suddenly stop responding to any commands that were directed to the DVR. We could watch any programming in real time, but everything that had been recorded just would not open when the icon for a DVRed show was clicked.
About the only thing that I figured I could try was to reset the system with the red button under the Hopper drop-down door. So I did this, and after re-acquiring satellites, testing switches and whatever else the box does, a satellite channel popped up on the screen AND... the DVR function was back to normal. AND, lo and behold, it's been working flawlessly for a couple of days now with NO trace of the picture pixelization/hesitation that I originally complained about. Sat signal levels show the same as they did when the problem was at its worst, so I don't think it was an RF-domain problem at all.
Thinking back on what we were doing before the DVR crashed, the only thing that could have been abnormal operation was that the Hopper had been left on Pause on a satellite signal for a very long time before we just shut the system down with the remote. Normally we'll do a Pause on what we're watching, or change channels, or do a playback Stop command, all just because it seems the proper thing to do. But it would seem to me that nothing a user could do in normal (or abnormal) operation could hose-up performance over the long haul; that is, create a hardware/firmware glitch that would hold over from day to day.
Anyhow, it's back to normal here, and what I've outlined is our experience... your mileage may vary. Many thanks to all who offered suggestions, I've still learned a lot without knowing what really happened.