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/discord.ts

@@ -5,6 +5,7 @@
  * @author Naomi Carrigan
  */
 /* eslint-disable @typescript-eslint/naming-convention -- Discord API requires snake_case fields and HTTP headers require Pascal-Case */
+import { logger } from "./logger.js";
 
 interface DiscordTokenResponse {
   access_token:  string;
@@ -50,18 +51,28 @@ const exchangeCode = async(
     redirect_uri:  redirectUri,
   });
 
-  const response = await fetch("https://discord.com/api/v10/oauth2/token", {
-    body:    parameters.toString(),
-    headers: { "Content-Type": "application/x-www-form-urlencoded" },
-    method:  "POST",
-  });
+  try {
+    const response = await fetch("https://discord.com/api/v10/oauth2/token", {
+      body:    parameters.toString(),
+      headers: { "Content-Type": "application/x-www-form-urlencoded" },
+      method:  "POST",
+    });
 
-  if (!response.ok) {
-    throw new Error(`Discord token exchange failed: ${response.statusText}`);
+    if (!response.ok) {
+      throw new Error(`Discord token exchange failed: ${response.statusText}`);
+    }
+
+    /* eslint-disable-next-line @typescript-eslint/consistent-type-assertions -- Response JSON matches DiscordTokenResponse shape */
+    return await (response.json() as Promise<DiscordTokenResponse>);
+  } catch (error) {
+    void logger.error(
+      "discord_exchange_code",
+      error instanceof Error
+        ? error
+        : new Error(String(error)),
+    );
+    throw error;
   }
-
-  /* eslint-disable-next-line @typescript-eslint/consistent-type-assertions -- Response JSON matches DiscordTokenResponse shape */
-  return await (response.json() as Promise<DiscordTokenResponse>);
 };
 
 /**
@@ -73,16 +84,26 @@ const exchangeCode = async(
 const fetchDiscordUser = async(
   accessToken: string,
 ): Promise<DiscordUser> => {
-  const response = await fetch("https://discord.com/api/v10/users/@me", {
-    headers: { Authorization: `Bearer ${accessToken}` },
-  });
+  try {
+    const response = await fetch("https://discord.com/api/v10/users/@me", {
+      headers: { Authorization: `Bearer ${accessToken}` },
+    });
 
-  if (!response.ok) {
-    throw new Error(`Discord user fetch failed: ${response.statusText}`);
+    if (!response.ok) {
+      throw new Error(`Discord user fetch failed: ${response.statusText}`);
+    }
+
+    /* eslint-disable-next-line @typescript-eslint/consistent-type-assertions -- Response JSON matches DiscordUser shape */
+    return await (response.json() as Promise<DiscordUser>);
+  } catch (error) {
+    void logger.error(
+      "discord_fetch_user",
+      error instanceof Error
+        ? error
+        : new Error(String(error)),
+    );
+    throw error;
   }
-
-  /* eslint-disable-next-line @typescript-eslint/consistent-type-assertions -- Response JSON matches DiscordUser shape */
-  return await (response.json() as Promise<DiscordUser>);
 };
 
 /**