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/game.spec.ts

@@ -420,6 +420,45 @@ describe("game route", () => {
     });
   });
 
+  describe("GET /load error path", () => {
+    it("returns 500 when the database throws during load", async () => {
+      vi.mocked(prisma.gameState.findUnique).mockRejectedValueOnce(new Error("DB error"));
+      vi.mocked(prisma.player.findUnique).mockRejectedValueOnce(new Error("DB error"));
+      const res = await app.fetch(new Request("http://localhost/game/load"));
+      expect(res.status).toBe(500);
+    });
+
+    it("returns 500 when the database throws a non-Error value during load", async () => {
+      vi.mocked(prisma.gameState.findUnique).mockRejectedValueOnce("raw string error");
+      vi.mocked(prisma.player.findUnique).mockRejectedValueOnce("raw string error");
+      const res = await app.fetch(new Request("http://localhost/game/load"));
+      expect(res.status).toBe(500);
+    });
+  });
+
+  describe("POST /save error path", () => {
+    const save = (body: Record<string, unknown>) =>
+      app.fetch(new Request("http://localhost/game/save", {
+        method: "POST",
+        headers: { "Content-Type": "application/json" },
+        body: JSON.stringify(body),
+      }));
+
+    it("returns 500 when the database throws during save", async () => {
+      const state = makeState();
+      vi.mocked(prisma.gameState.findUnique).mockRejectedValueOnce(new Error("DB error"));
+      const res = await save({ state });
+      expect(res.status).toBe(500);
+    });
+
+    it("returns 500 when the database throws a non-Error value during save", async () => {
+      const state = makeState();
+      vi.mocked(prisma.gameState.findUnique).mockRejectedValueOnce("raw string error");
+      const res = await save({ state });
+      expect(res.status).toBe(500);
+    });
+  });
+
   describe("POST /reset", () => {
     const reset = () =>
       app.fetch(new Request("http://localhost/game/reset", { method: "POST" }));
@@ -450,5 +489,17 @@ describe("game route", () => {
       const body = await res.json() as { signature: string | undefined };
       expect(typeof body.signature).toBe("string");
     });
+
+    it("returns 500 when the database throws during reset", async () => {
+      vi.mocked(prisma.player.findUnique).mockRejectedValueOnce(new Error("DB error"));
+      const res = await reset();
+      expect(res.status).toBe(500);
+    });
+
+    it("returns 500 when the database throws a non-Error value during reset", async () => {
+      vi.mocked(prisma.player.findUnique).mockRejectedValueOnce("raw string error");
+      const res = await reset();
+      expect(res.status).toBe(500);
+    });
   });
 });