Sublime Text Ctrl C copy
Update: I'm still not sure what the problem was, but deleting ~/Library/Application Support/Sublime Text 3 solved it. It looks like my default Sublime packages got screwed up to the point where it was somehow overriding all the user provided keymaps, among other issues. But now that I've regenerated the folder, setting keybindings works like normal.

As for why only Ctrl+C was affected and not Ctrl Z, X, or V, it's because Sublime has a default shortcut for Ctrl+C defined on Macs (cancel_build), but no shortcuts for Ctrl+Z, X, or V. Other shortcuts which clash with a predefined shortcut such as Ctrl+K were also broken.