Bryce,
I can't see that Premiere Elements is not doing well!
This is a screen grab from your Canon footage. It may be that with all the subject motion, focus, detail and panning that the camera is not getting it "picture perfect". Maybe the bitrate is a little low..
Comparing that to a screen grab from you completed video, I don't see much change in picture quality. Maybe some, but certainly not in the "sucks" category.

Later in your video you have shot the owners and their dog. The sky is blown out, but detail in the couple and dog are very good. There is nothing to suggest that your camera or Premiere Elements is not functioning well.

Regarding the loss of quality during the editing process, that can be different. For editing, preview files are being created for you under the hood. There are a couple setting that effect that, but it is a preview quality from "real time" rendering. At final output all the source files are re-read for the encoding process. Consequently, it is possible to have less than the best previews during editing but great quality at final output.
Good luck and have fun with your video editing projects.
Bill