Docs: refresh widget code copies and document the wallpaper failure toast

[SandObserver]

Summary

• Resync public/js/ from the app repository, picking up the badge ink rule in badge-logic.js and its cache-busting reference in dashboard.js.
• Add a "Messages you might see" row for the wallpaper failure toast, naming the unreachable link, unaccepted format and 16 MB size causes.

Testing
All green: npm run build and npm run check both pass. Widget previews on all eleven widget pages and the landing page render populated content in Chrome, in both colour schemes, with no console errors.

Checklist

• npm run build and npm run check both pass.
• Product UI is shown by running or capturing it, never recreated in CSS.
• Any new preview or screenshot carries no private hostnames or real data.
• Visual changes verified in both colour schemes and both docs themes.
• Links to the app repo still resolve.

[cloudflare-workers-and-pages]

Deploying stackyard-docs with    Cloudflare Pages

Latest commit:	391a634
Status:	✅  Deploy successful!
Preview URL:	https://19bb25c2.stackyard-docs.pages.dev
Branch Preview URL:	https://docs-sync-1-8-0.stackyard-docs.pages.dev

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