Question! Is there any sort of mod that collectively allows all races to use all eyes available? Or would I have to individually add them through some sort of .esp? I recently downloaded the "cataract eyes" mod from TESNexus (http://tesnexus.com/downloads/file.php?id=35519) and had to add them via editing some random .esp that was low on my load order so I could use them. I'm not sure this is the right way to go about things... Or is there some sort of external program I can use to switch my eyes around?

Two words - Wrye Bash

Does Wrye Bash allow all races to use all eyes, though? I've only gotten it to allow me to use various mods that add new eyes to work at the same time. I mean to say, is there a way to make Mystic Elf eyes work on Bretons (or something like that)? Sorry if I'm being unclear!

In this instance all you've added is textures to your data file folder.
To make use of these you would have to build a mod then merge them into your game with Bash.
Capucines Eye Expansion would give you something similar without the trouble.