Can you tell us more about that original AVI file, and also the Sequence settings, where you are editing?
This is an unconventional way or working, but I don't see why it shouldn't work. You just right click your clip and select New Sequence from Clip to create a sequence that matches the settings of your original clip.
Then you edit out all of the parts you don't want, export it and re-import it. If there are still issues it would help to see screenshots of your clip's specifications, sequence settings and export settings.
save the file back to itself with only the good parts.
You can't. PP will only create a new file when exporting. It won't overwrite the very file it's reading from to create the export.
But there is no need to create an export if you're not done editing. Cutting out the bad parts is the very essence of what this software and editing in general is for. You just need to first learn how to use it.
I agree with Jim. Most editors keep all their footage files in the project panel. Then they double click a clip to play it in the source viewer. Then they scrub through the clip using the J, K, L keys, set in and out points using the I & O keys and finally drag the trimmed clip onto the timeline. This is how most editors work and I think you might be surprised at how efficient it is. Most editors try not to export their footage more than necessary because it looses quality each time it's rendered it out. As jim said, this is exactly what editing software is good for. Check out this tutorial for some good CS6 workflow suggestions.