feat: refreshing UX/UI

[gnugomez]

Related to #1934

Beyond the visual redesign of the search page and overall layout, I'm introducing a few new features:

• A real home page. The landing page is no longer the search results — it's a hero search with popular-search chips, a browse-by-category grid, curated extension rows (most downloaded, recently updated) and get-involved cards. Everything is data-driven through PageSettings.elements.home, and deployments can replace the page entirely; I export the building blocks (HeroSearch, BrowseCategories, CuratedSections, GetInvolved, ExtensionCard) so custom home pages can be composed from them.
• A dedicated search page. Search now lives at /search, with the query, category and sorting reflected in the URL so searches are shareable and survive reloads. Old /?search=… URLs redirect automatically. Categories filter from a sticky sidebar on desktop and a scrollable pill row on mobile.
• Keyboard shortcuts. Pressing ? opens a shortcuts modal. / focuses the search field from anywhere, h/s jump to home/search, d opens the docs, p goes to publishing, and o/c/r switch tabs on an extension page.
• Keyboard-driven results. The search input and the results grid share a cursor: ↑/↓ step through results while the caret stays in the field, Enter opens the highlighted extension.
• New footer The footer is now built from data — brand, link columns, social icons, copyright — via PageSettings.elements.footer, or can be swapped for a custom component. The legacy footer API keeps working.
• UX improvements The hero search morphs into the navbar field using the View Transitions API, forward navigations scroll to the top without breaking back/forward restoration, and focus outlines are visible across all the new controls.

Assisted-By: anthropic:claude-sonnet-4-6, anthropic:claude-fable-5

[netomi]

I did compact the user avatar component to avoid having separate profile and settings items which is confusing.

The mobile menu part needs improvement imho, I do not like the current way with nested menus.

[netomi]

LGTM, merge as is and work on some improvements in other PRs so we can base work on that.