Well that was an interesting test. It was interesting to see parts of the animation being rendered, but unfortunately it didn't fix the problem.
Thanks for the idea.
I did find a small help. I noticed that the problem is related to how many frames I render. I'm not talking about a lot though. My entire test animation was only 133 frames. But when I rendered just the problem area, about 20 frames, the problem went away. Not a great fix (who wants to render 20 frames at a time

)
I thought turning of layer effects might help, but, surprisingly to me, it didn't.
Just Found Something that Works Pretty Well
I did just try something else. I used my Corel Video Studio to input AVI and output MP4. That worked out very well. The result is only 2x the ASP mp4 output, but the quality is much better and it's from 10x to 300x smaller thant the AVI. I love Video Studio! It's the best I've ever seen of easily dealing with all the different video formats. It just seems to work and easy to tweak.
That works, but it's too bad that, as of yet, in some cases, I can't just render straight out of ASP with good quality
I'm still interested any any ideas to output better quality with small file size if people have other ideas.