part 1 Assume that I sampled a recording of a Nirvana song, a recording of Bernstein's Westside Story and a recording of Hold my Hand by the Beatles. Then I use bits and pieces of each to post a new recording. Further assume that you can't tell from whence the bits and pieces originate once I (complete) my piece. Do I need the permission of anyone to do that? If so, what permission do I need?

You can check it out the two cases that have handled this issue. This means that the interpretation of this law is not settled. So just explained which court's opinion you are following when you address this question.

This Court ruled no permission needed -

https://fairuse.stanford.edu/case/vmg-salsoul-llc-v-ciccone/ This court ruled permission needed - https://en.wikipedia.org/wiki/Bridgeport_Music,_Inc._v._Dimension_Films

Part 2

Does fair use cover this YouTube video owner's use of other videos?

https://youtu.be/tprMEs-zfQA

https://youtu.be/EsBfj6khrG4 Use one argument from here: https://web.law.duke.edu/cspd/comics/