The "Rewrite with AI" buttons in the editor are greyed out. Do I need a Perplexity key?

No. The rewrite buttons (improve, shorten, expand and so on) switch on as soon as you select some text in the editor: click and drag over a passage first. They use the AI engine of your book and need no extra key. The Perplexity key mentioned lower in the same panel is optional and is only used by the fact-check feature; nothing else in the editor needs it.