[ko] Translate shared website navigation and search UI into Korean #880

Goal

Translate the shared website shell into Korean while keeping all English routes unchanged. Page content is assigned in separate translation issues.

Scope and source

• apps/landing/src/components/Header/: desktop/mobile navigation, search placeholder, theme/menu button accessible names.
• apps/landing/src/app/(detail)/docs/LeftMenu.tsx and components/LeftMenu.tsx: shared sidebar labels, groups and overview labels; keep API component names.
• apps/landing/src/app/(detail)/docs/RightIndex.tsx: Contents and Edit this page.
• apps/landing/src/components/SearchModal/: loading, empty state and input/button labels.
• apps/landing/src/components/Footer.tsx: remaining English text and image alt text; retain existing Korean legal details.

Work

Use next-intl useTranslations and add matching keys to apps/landing/src/i18n/messages/en.json and ko.json. Pass the English catalog (English pages) or a merge of English defaults and Korean translations (Korean pages) through NextIntlClientProvider in SiteLayout. The current infrastructure provides locale context but intentionally leaves UI copy English; this issue introduces the catalogs. Add translations for visible labels and attributes as needed. Preserve URLs, filter/state values and component/API identifiers. Do not translate document bodies or home-page marketing content in this issue.
Catalog wiring: load the English catalog and merge Korean overrides per namespace in src/i18n/request.ts, then pass getMessages({locale}) to NextIntlClientProvider in SiteLayout. Client components use useTranslations; server components use getTranslations({locale}) with an explicit locale.

Getting started

This is intended for a first-time contributor. Comment to claim this page before starting, and send one focused PR. Questions are welcome.

Prerequisite: #881. Wait for that infrastructure to merge before submitting the translation. Context: #584.

Read the translation guide and use the Korean overview at apps/landing/src/app/(detail)/docs/overview/page.ko.mdx as an example. English stays at the original URL; untranslated /ko URLs automatically redirect to their English page.

Acceptance checklist

• Translate the complete assigned scope into natural Korean, including headings, prose, table descriptions and metadata.
• Keep API/package names, code behavior, numeric data, external links and benchmark caveats unchanged. Explanatory code comments may be translated.
• Preserve MDX imports and JSX structure. Fix relative imports when moving files; prefer @/ imports.
• Keep internal links root-relative (for example /docs/installation); locale-aware navigation adds /ko automatically.
• Verify both English and Korean URLs, mobile and desktop, links, tables and code blocks.
• From the repository root run bun install and bun run build; from apps/landing run bun dev and bun run build. Run node script.js to refresh search data; translated pages are connected explicitly in src/i18n/pages.ts. Run the type, lint and i18n checks listed in the translation guide.
• Keep the English copy unchanged. Do not translate other pages in this PR.