This is not correct; you're simply required to follow all applicable licenses at the same time. This may or may not be possible, but is in practice quite commonly done.
Completely depends on how much you've "mixed in", and facts specific to that individual work.
Fair use doesn't get thrown out the window because GPL authors have a certain worldview.
Second, there are a lot of non-copyrightable components to source code - if you can't copyright it - you certainly can't GPL it. These can be copied freely by anyone at any time.
You should keep them in different directories and have the appropriate license for each directory. You can have a top-level LICENSE file explaining the situation.