Does the Junk Yard curtain need Accessibility permission or steal focus?
Short answer: No. The curtain doesn't need Accessibility permission and doesn't take focus, so your current app stays in front and ⌘V pastes straight into it.
No to both. The curtain doesn't ask for Accessibility permission, and it doesn't take focus — the app you were working in stays in front while the curtain is open. That's why you can click a tile and then simply press ⌘V where you were typing.