For any help, you'll need to supply more details including, computer, PrE version, the project setting, how it was set, what the camera recording settings are (or their file specifications) and how you are recording your audio.
At first glance, it appears that the project settings, camera settings and output setting are a mis-match.
Are you saying that there are no issues at all with the Nikon D5000 which appears to be offering you choices of
1280 x 720 @ 24 progressive frames per second or 640 x 424 @ 24 progressive frames per second or 320 x 216 @ 24 progressive frames per second. Which of those 3 resolutions are you using? It is using MotionJPEG as the video compression to go along with the .avi file extension. Typically the user has problems with the MotionJPEG video compression which you are not in any way. Please confirm.
You say that your problems are with the .mov video from your friend's camcorder. What is the frame size and frame rate of this .mov video? What is the video compression that goes along with the .mov file extension? Quickest way to get this type of information is to supply the brand/model/settings of the camcorder that recorded these .mov files.
Major focus points here
a. What version of Premiere Elements and what computer operating system involved?
b. Do you have the latest version of QuickTime installed on your computer with Premiere Elements?
c. What is the project preset for this project that is apparently using more than one format?
The project preset is supposed to match the properties of the source media. When there is more than one format, then you need to set priorities and fit everything else into it. If you let the project set the project preset, the project preset in your case could reflect the .avi or .mov, whichever you dragged to the Timeline first.
In this regard, please tell us the readings for Editing Mode, Timebase, Frame Size, and Pixel Aspect Ratio (even if grayed out). See Edit Menu/Project Settings/General.
d. Are you saying that the problem exists before and after export of the file saved to the computer hard drive?
What export settings did you use to create the YouTube video that you posted as demo for the problem? Is what is shown in your YouTube video what you are seeing before and after Timeline content?
Please review and consider. If any questions or need clarification, please do not hesitate to ask. We can go at whatever troubleshooting pace you prefer.
We look forward to your details which will help us to help you.