feat: add periodic game.status broadcast and live status REST endpoint

Add 20-second game.status WebSocket heartbeat from active shard monitors
containing full game state, and GET /status-live REST endpoint for on-demand
polling. Fix missing token destructuring in SessionInfo causing crash.
Relax frontend polling from 3s to 60s since WebSocket events now cover
real-time updates. Bump version to 0.6.0.

Made-with: Cursor

docs/api/websocket.md

@@ -136,6 +136,7 @@ Must be authenticated.
 | `game.started` | Game transitioned to Gameplay (broadcast to subscribers) |
 | `game.ended` | Game finished (broadcast to subscribers) |
 | `room.disconnected` | Shard lost connection to Jackbox room (broadcast to subscribers) |
+| `game.status` | Periodic game state heartbeat every 20s (broadcast to subscribers) |
 | `player-count.updated` | Manual player count override (broadcast to subscribers) |
 | `vote.received`        | Live vote recorded (broadcast to subscribers)   |
 
@@ -318,6 +319,29 @@ All server-sent events use this envelope:
 
 Possible `reason` values: `room_closed`, `room_not_found`, `connection_failed`, `role_rejected`, `manually_stopped`.
 
+### game.status
+
+- **Broadcast to:** Clients subscribed to the session
+- **Triggered by:** Periodic 20-second heartbeat from an active shard monitor. Also available on demand via `GET /api/sessions/{sessionId}/games/{sessionGameId}/status-live`.
+
+**Data:**
+```json
+{
+  "sessionId": 1,
+  "gameId": 5,
+  "roomCode": "LSBN",
+  "appTag": "drawful2international",
+  "maxPlayers": 8,
+  "playerCount": 4,
+  "players": ["Alice", "Bob", "Charlie", "Diana"],
+  "lobbyState": "CanStart",
+  "gameState": "Lobby",
+  "gameStarted": false,
+  "gameFinished": false,
+  "monitoring": true
+}
+```
+
 ### player-count.updated
 
 - **Broadcast to:** Clients subscribed to the session
@@ -536,6 +560,10 @@ ws.onmessage = (event) => {
       console.log('Room disconnected:', msg.data.reason);
       break;
 
+    case 'game.status':
+      console.log('Status heartbeat:', msg.data.roomCode, '- players:', msg.data.playerCount, '- state:', msg.data.gameState);
+      break;
+
     case 'player-count.updated':
       console.log('Player count:', msg.data.playerCount, 'for game', msg.data.gameId);
       break;