feat: poll countdown timer, game-selection sync, source tracking, and multi-admin fixes

Work spanning May 7-10 across multiple sessions:

Poll winner detection + source column (May 7):
- Fix race condition in handleEndPolling where WS voting.ended cleared
  leadingGame before the setTimeout could capture the winner
- Add pollActiveRef guard to prevent late poll.leading messages from
  re-activating an ended poll
- Add 'source' column to session_games (dice/manual/poll) with backward-
  compatible fallback from manually_added flag
- Show indigo "Poll" badge in game lists (Picker, Home, SessionDetail)
- Include source in session export (JSON and text formats)

Multi-admin poll state sync (May 9):
- Enrich poll.start broadcast with pollStartedAt timestamp so all admin
  clients can start their timers from the correct time
- Enrich voting.ended broadcast with winnerGameId/Label/Votes so all
  admins see the winner prompt, not just the one who clicked End Poll
- Add poll.start WS handler in SessionInfo so Admin B sees polls started
  by Admin A without refreshing
- Make handleStartPolling optimistic with rollback on failure

WebSocket keepalive + auto-reconnect (May 9):
- Add 30s ping interval to SessionInfo WS connection (matching server's
  60s timeout) to prevent silent disconnects
- Add auto-reconnect on close with 3s delay
- Proper cleanup of ping interval, reconnect timeout, and onclose handler

Sync selected game across admin clients (May 10):
- New POST/DELETE /sessions/:id/game-selection endpoints with DB
  persistence (pending_game_id, pending_game_source columns)
- Broadcast game.picked/game.dismissed WS events to session subscribers
- handleDismissGame replaces inline setSelectedGame(null) calls
- Restore pending game selection on page load for late-joining admins
- Clear pending selection when game is formally added to session

Poll ending countdown timer (May 10):
- POST /:id/voting/end now accepts optional { delay } (0-300 seconds)
- New POST /:id/voting/cancel-end to abort a scheduled end
- New poll.ending and poll.ending.cancelled WS events
- poll_ending_at column on sessions table for crash recovery
- rescheduleEndingPolls() called on server startup to resume countdowns
- End Poll button opens popover with End Now / 5s / 10s / 30s / custom
- Red "Poll Ending" card with countdown display and Cancel button
- Document new WS events in docs/api/websocket.md

Co-authored-by: Cursor <cursoragent@cursor.com>

backend/server.js

@@ -22,6 +22,7 @@ app.get('/health', (req, res) => {
 const authRoutes = require('./routes/auth');
 const gamesRoutes = require('./routes/games');
 const sessionsRoutes = require('./routes/sessions');
+const { rescheduleEndingPolls } = require('./routes/sessions');
 const statsRoutes = require('./routes/stats');
 const pickerRoutes = require('./routes/picker');
 const votesRoutes = require('./routes/votes');
@@ -54,6 +55,7 @@ if (require.main === module) {
   server.listen(PORT, '0.0.0.0', () => {
     console.log(`Server is running on port ${PORT}`);
     console.log(`WebSocket server available at ws://localhost:${PORT}/api/sessions/live`);
+    rescheduleEndingPolls();
   });
 
   const shutdown = async () => {