Is there a way to edit the audio of a source clip and have those changes reflect in the timeline clips?
In other words, I made the edits and subsequently learned that the audio could actually be cleaned up a bit better than I'd done it. But I'd like to run filters in the source clips instead of having to go into the timeline to clean the audio up cut by cut. Is it possible to clean the source files and have the changes just cascade to the clips on the timeline?
It depends on how you do the work. If the work is done in an outside program (Audition, for example) and saved into the file itself, then yes, when you reopen PP the new audio will be heard for each clip instance.
But there's no way to do that entirely within PP. There's no way to apply an effect to a clip except when it's in a sequence, and those effects do not ripple throughout the project.