How do I preserve file paths after cloning a failing external drive?
If by cloning you mean you're going to use software such as Carbon Copy Cloner or SuperDuper! to create an exact image of your external drive, and as long as you ensure the external drive maintains the same volume name, then you should have no problems.

Once you've made the copy, however, I would no longer keep the other external drive connected.

Libraries: iTunes and Photos

In your question you state:


  How do I make sure that, after cloning the drive, my applications will
  read their files from the new drive without me manually having to
  re-specify the file locations? I.E. not having to manually update the
  file locations of every song in my iTunes library.


Well, if you're only concerned about your Photos and iTunes library, then even if you didn't clone your external hard drive, you'd still be fine as you can manually set where the iTunes Library and Photos Library are located. This is a once off step. For example, once iTunes knows where it's Library is, it will find everything. You don't need to manually update the file locations of every song in your iTunes library.