feat: one-click join + live pod presence

Join UX: click a member's name to join directly, or type a name and "Join"
(adds to roster + joins) — removes the redundant dropdown step. "Add" still
adds to the roster without joining.

Presence: new GET /api/presence (LiveKit RoomServiceClient) reports who's
connected per pod. The pod list polls it and shows a "N in room" badge plus a
green dot on members currently connected, so people can see active pods and jump in.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

backend/src/livekit/rooms.ts

@@ -0,0 +1,39 @@
+import { RoomServiceClient } from 'livekit-server-sdk';
+import { env } from '../env.js';
+
+function isConfigured(): boolean {
+  return !!env.LIVEKIT_URL && !env.LIVEKIT_URL.includes('REPLACE_ME');
+}
+
+let client: RoomServiceClient | null = null;
+function svc(): RoomServiceClient {
+  if (!client) {
+    const httpUrl = env.LIVEKIT_URL.replace(/^wss:/, 'https:').replace(/^ws:/, 'http:');
+    client = new RoomServiceClient(httpUrl, env.LIVEKIT_API_KEY, env.LIVEKIT_API_SECRET);
+  }
+  return client;
+}
+
+/**
+ * Who is currently connected in each pod's LiveKit room, as display names,
+ * keyed by pod id (= room name). Empty rooms are omitted. Returns {} if
+ * LiveKit isn't configured or the API call fails (presence is best-effort).
+ */
+export async function getPresence(): Promise<Record<string, string[]>> {
+  if (!isConfigured()) return {};
+  const out: Record<string, string[]> = {};
+  const rooms = await svc().listRooms();
+  await Promise.all(
+    rooms
+      .filter((r) => r.numParticipants > 0)
+      .map(async (r) => {
+        try {
+          const ps = await svc().listParticipants(r.name);
+          out[r.name] = ps.map((p) => p.name || p.identity);
+        } catch {
+          out[r.name] = [];
+        }
+      }),
+  );
+  return out;
+}