The problem of my top thread not catching the bottom thread has been solved . . . I think. I changed the needle! When I took off the old needle, the poor thing looked like it had been through the wringer. Not only was it dull, but it was also bent and dinged in several places! I have no idea how it got so bent out of shape, but as soon as I put the new needle in, it stitched just fine.
Now my actual free-motion quilting is another story. It's bad. Pitiful. Sad.