Decouple room monitoring from player count, fix Jackbox API fetch

Extracts checkRoomStatus into shared jackbox-api.js with proper
User-Agent header (bare fetch was silently rejected by Jackbox API)
and always-on error logging (previously gated behind DEBUG flag).

Splits room-start detection (room-monitor.js) from audience-based
player counting (player-count-checker.js) to eliminate circular
dependency and allow immediate game.started detection. Room monitor
now polls immediately instead of waiting 10 seconds for first check.

Made-with: Cursor

backend/utils/jackbox-api.js

@@ -0,0 +1,42 @@
+const JACKBOX_API_BASE = 'https://ecast.jackboxgames.com/api/v2';
+
+const DEFAULT_HEADERS = {
+  'User-Agent': 'Mozilla/5.0 (compatible; GamePicker/1.0)'
+};
+
+/**
+ * Check room status via the Jackbox ecast REST API.
+ * Shared by room-monitor (polling for lock) and player-count-checker (room existence).
+ */
+async function checkRoomStatus(roomCode) {
+  try {
+    const response = await fetch(`${JACKBOX_API_BASE}/rooms/${roomCode}`, {
+      headers: DEFAULT_HEADERS
+    });
+
+    if (!response.ok) {
+      console.log(`[Jackbox API] Room ${roomCode}: HTTP ${response.status}`);
+      return { exists: false };
+    }
+
+    const data = await response.json();
+    const roomData = data.body || data;
+
+    if (process.env.DEBUG === 'true') {
+      console.log('[Jackbox API] Room data:', JSON.stringify(roomData, null, 2));
+    }
+
+    return {
+      exists: true,
+      locked: roomData.locked || false,
+      full: roomData.full || false,
+      maxPlayers: roomData.maxPlayers || 8,
+      minPlayers: roomData.minPlayers || 0
+    };
+  } catch (e) {
+    console.error(`[Jackbox API] Error checking room ${roomCode}:`, e.message);
+    return { exists: false };
+  }
+}
+
+module.exports = { checkRoomStatus };