refactor(docs): extract remaining components and add lazy loading

Extract Nav, Footer, PageShell, PrevNextNav, MobilePicker,
SidebarLLMButton, and CopyButton into separate component files.
Move GuidePage, ReferencePage, and RawGuidePage into pages/ directory.
Add React.lazy() for all page-level components to enable code splitting.
Extract SearchContext into shared lib module. App.tsx reduced from
~1,055 lines to ~60 lines.