It is my understanding that you can easily download video (your own) that you have uploaded to YouTube without issue if you lost track of the original copy of the file that you uploaded to YouTube in the first place. That can be done at the You Tube web site.
But, if you are dealing with the YouTube video of others, then I would refer you to a description of the video at the YouTube site as well as the YouTube Agreement.
Whatever project that you are doing with your own source media, please remember that .AVI is just a file extension that can have different video and audio compression. So, you need to know what type of .avi file you have as well as the rest of the video properties and relate them to establishing the appropriate Premiere Elements 10 Windows project.
For more background on AVI as a "wrapper," this FAQ Entry will provide details: http://forums.adobe.com/thread/440037?tstart=0
Also, by the time that a video gets onto YouTube, it has probably undergone at least two, and possibly more, compression stages. When one downloads, or does a video screen-cap of the material on YouTube, it is likely that at least one more compression step will take place. Then, after Import and editing, and depending on output, more compression will very likely be applied. The material then basically will be equal to a FAX, of a FAX, of a FAX, of a FAX, of a FAX - i.e. not very good, regarding quality. Just something to think about.