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/test/routes/leaderboards.spec.ts

@@ -152,6 +152,18 @@ describe("leaderboards route", () => {
     expect(typeof body.entries[0]?.activeTitle).toBe("string");
   });
 
+  it("returns 500 when the database throws", async () => {
+    vi.mocked(prisma.player.findMany).mockRejectedValueOnce(new Error("DB error"));
+    const res = await get();
+    expect(res.status).toBe(500);
+  });
+
+  it("returns 500 when the database throws a non-Error value", async () => {
+    vi.mocked(prisma.player.findMany).mockRejectedValueOnce("raw string error");
+    const res = await get();
+    expect(res.status).toBe(500);
+  });
+
   it("defaults to 0 for game-state categories when state is missing", async () => {
     vi.mocked(prisma.player.findMany).mockResolvedValueOnce([makePlayer({ discordId: "p1" })] as never);
     vi.mocked(prisma.gameState.findMany).mockResolvedValueOnce([] as never);