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>

apps/api/src/services/webhook.ts

@@ -5,6 +5,8 @@
  * @author Naomi Carrigan
  */
 /* eslint-disable @typescript-eslint/naming-convention -- Discord API requires snake_case and Pascal-case header names */
+import { logger } from "./logger.js";
+
 const discordApi = "https://discord.com/api/v10";
 
 /**
@@ -34,7 +36,13 @@ const grantApotheosisRole = async(discordId: string): Promise<void> => {
         method:  "PUT",
       },
     );
-  } catch {
+  } catch (error) {
+    void logger.error(
+      "webhook_apotheosis_role",
+      error instanceof Error
+        ? error
+        : new Error(String(error)),
+    );
     // Graceful degradation — role grant failure must not affect the apotheosis
   }
 };
@@ -81,7 +89,13 @@ const postMilestoneWebhook = async(
       headers: { "Content-Type": "application/json" },
       method:  "POST",
     });
-  } catch {
+  } catch (error) {
+    void logger.error(
+      "webhook_milestone",
+      error instanceof Error
+        ? error
+        : new Error(String(error)),
+    );
     // Graceful degradation — webhook failure must not affect the game action
   }
 };