Garage Door: Fixed!
I fiddled with the garage door opener for about 30 minutes before I knew what needed to be done to fix it. I had to open the cover on the opener to see how it works. When I did that, I found that there is a pair of limit switches that stop the door at the right location when it is approaching fully-closed and when it is approaching fully-open. Those switches are mounted to a plate which mounts to the motor. The plate-motor mount was loose, and, as a result, the limit switches were no longer doing the right thing. The door didn't open all the way any more, and the motor continued to grind after the door closed. Not good.
To fix the problem, I first secured the limit switch mount plate to the motor by tightening the nuts as securely as possible. Now, at least, it wouldn't flop around as the door opened or closed. Then, I had to re-adjust the limit switches. I pulled the vertical plate to the right and moved the large white nuts on the screw to change the location of the cutoff points. By adjusting the location of these nuts, you effectively change when the motor turns on and off and how far the door opens and closes.
Once I knew what to do and found the right tools, it took about 15 minutes. But, I sunk lots of time trying to get to that point. Oh well. It's good to have this working again.