feat: error handling, logging, analytics, OG tags, and sticky sidebar (#44)

## Summary

- Add comprehensive try/catch error handling across all API routes, middleware, and the Hono global error handler, piping every unhandled error to the `@nhcarrigan/logger` service to prevent silent crashes and unhandled Promise rejections
- Add a `logError` utility on the frontend that forwards errors through the overridden `console.error` to the backend telemetry endpoint; apply it to every silent `catch {}` block in the game context, sound, notification, and clipboard utilities, and wrap the React tree in an `ErrorBoundary`
- Add Plausible analytics, Open Graph + Twitter Card meta tags, Tree-Nation widget, and Google Ads to `index.html`
- Make the game sidebar sticky with a `--resource-bar-height` CSS custom property offset so it stays viewport-height without overlapping the resource bar; reset sticky behaviour in the mobile responsive override

## Test plan

- [ ] Lint passes: `pnpm lint`
- [ ] Build passes: `pnpm build`
- [ ] Verify errors thrown in API routes appear in the logger service rather than crashing the process
- [ ] Verify frontend errors appear in the `/api/fe/error` backend log
- [ ] Verify Open Graph tags render correctly when sharing the URL
- [ ] Verify Plausible analytics fires on page load
- [ ] Verify Tree-Nation badge renders in the sidebar
- [ ] Verify sidebar stays fixed while the main content scrolls on desktop
- [ ] Verify mobile layout is unaffected

✨ This issue was created with help from Hikari~ 🌸

Reviewed-on: #44
Co-authored-by: Hikari <hikari@nhcarrigan.com>
Co-committed-by: Hikari <hikari@nhcarrigan.com>

pnpm-lock.yaml

@@ -23,6 +23,9 @@ importers:
       '@hono/node-server':
         specifier: 1.13.7
         version: 1.13.7(hono@4.7.4)
+      '@nhcarrigan/logger':
+        specifier: 1.1.1
+        version: 1.1.1
       '@prisma/client':
         specifier: 6.5.0
         version: 6.5.0(prisma@6.5.0(typescript@5.8.2))(typescript@5.8.2)
@@ -689,6 +692,9 @@ packages:
       typescript: '>=5'
       vitest: '>=2'
 
+  '@nhcarrigan/logger@1.1.1':
+    resolution: {integrity: sha512-P6OEQFHDtf6psybYGljuCxkSW6DLQCsx1aZZ3w4YKBXHBFjDbhuvpM9K1kPhVN48hakitx2WPLEoIFr6YZELYw==}
+
   '@nhcarrigan/typescript-config@4.0.0':
     resolution: {integrity: sha512-969HVha7A/Sg77fuMwOm6p14a+7C5iE6g55OD71srqwKIgksQl+Ex/hAI/pyzTQFDQ/FBJbpnHlR4Ov25QV/rw==}
     engines: {node: '20', pnpm: '9'}
@@ -3490,6 +3496,8 @@ snapshots:
       - eslint-import-resolver-webpack
       - supports-color
 
+  '@nhcarrigan/logger@1.1.1': {}
+
   '@nhcarrigan/typescript-config@4.0.0(typescript@5.8.2)':
     dependencies:
       typescript: 5.8.2