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/routes/auth.ts

@@ -15,6 +15,7 @@ import {
   fetchDiscordUser,
 } from "../services/discord.js";
 import { signToken } from "../services/jwt.js";
+import { logger } from "../services/logger.js";
 import type { Player } from "@elysium/types";
 
 const authRouter = new Hono();
@@ -92,6 +93,8 @@ authRouter.get("/callback", async(context) => {
       });
 
       const jwtToken = signToken(player.discordId);
+      void logger.log("info", `New player registered: ${player.discordId}`);
+      void logger.metric("user_registered", 1, { discordId: player.discordId });
 
       // eslint-disable-next-line capitalized-comments -- v8 ignore
       /* v8 ignore next -- @preserve */
@@ -111,6 +114,8 @@ authRouter.get("/callback", async(context) => {
     });
 
     const jwtToken = signToken(updated.discordId);
+    void logger.log("info", `Player logged in: ${updated.discordId}`);
+    void logger.metric("user_login", 1, { discordId: updated.discordId });
 
     // eslint-disable-next-line capitalized-comments -- v8 ignore
     /* v8 ignore next -- @preserve */
@@ -118,7 +123,13 @@ authRouter.get("/callback", async(context) => {
     return context.redirect(
       `${clientUrl}/auth/callback?token=${jwtToken}&isNew=false`,
     );
-  } catch {
+  } catch (error) {
+    void logger.error(
+      "auth_callback",
+      error instanceof Error
+        ? error
+        : new Error(String(error)),
+    );
     // eslint-disable-next-line capitalized-comments -- v8 ignore
     /* v8 ignore next -- @preserve */
     const clientUrl = process.env.CORS_ORIGIN ?? "http://localhost:5173";