refactor: SearchBox.js is 2692 lines — extract subcomponents for maintainability

## Problem

`packages/web/src/components/search/SearchBox.js` is **2,692 lines** handling at least 6 distinct responsibilities:

1. Search input management (value, focus, blur, keyboard)
2. Suggestion rendering (8+ types: popular, recent, FAQ, featured, document, index, endpoint, predictive)
3. AI Answer mode (full chat UI with session management)
4. Voice/mic input
5. Image search with camera
6. URL param sync

The render method alone spans hundreds of lines with deeply nested ternaries.

## Suggested Fix

Extract these as additive subcomponents (non-breaking, each can be a separate PR):

1. **`AISection.js`** — The entire AI answer screen (~L1870-1988)
2. **`SuggestionsSection.js`** — All 8 suggestion types rendering (~L1990-2180)
3. **`SearchInput.js`** — The input element with mic, camera, and autofill addons

Each extraction is verifiable by existing snapshot tests since the composite output is unchanged.

## Severity

High — maintainability bottleneck

Start by reading packages/web/src/components/search/SearchBox.js and reviewing the existing snapshot tests. Work on one suggested extraction at a time—AISection.js, SuggestionsSection.js, or SearchInput.js—using the indicated responsibilities and line ranges as boundaries. Done means the composite output remains unchanged and the snapshot tests still pass.

Written by the indexing model from the issue text.