[Website] Move Playground tools into a bottom dock

[adamziel]

What changed

• Replaces the desktop Site Manager side panel with a stable bottom dock and centered pane.
• Moves Playgrounds, New, Settings, Current Blueprint, Database, Files, Logs, and Share/export into dock destinations.
• Keeps current Playground management lightweight: compact facts in Your Playgrounds, while tool flows remain in the dock.
• Preserves GitHub import/export, PR previews, Blueprint URL/gallery starts, .zip import/export, saved/recent Playgrounds, settings, database, files, logs, and current Blueprint editing.

Validation

• npm exec nx -- run playground-website:typecheck
• npm exec nx -- run playground-website:lint
• npm exec nx -- run playground-website:test
• npm exec nx -- run playground-website:build:standalone:production
• Manual browser check: opened Playgrounds, New, Settings, Current Blueprint, Database, Files, Logs, and Share; verified pane stability, preview blur/click-dismiss, and no JS console errors.