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My JavaScripts got broken! My cookies just don't work! [Insert error here]

If your javascript doesn't work anymore, best bet is to uncheck "External JavaScript includes" from "Using CDN on these contents" option section. You can also enable Debug mode to see which files get involved, and try excluding them one by one to find out the problematic one and deal with it for good.

Why doesn't the plugin parse the external stylesheet and rewrite the background images there too?

There should be no need. If the stylesheet got rewritten and is served through a destination host, all relative (most commonly) background images defined inside it will be pulled from that host too.