Expose Prometheus metrics on a separate port

Metrics are served on METRICS_PORT (default 2112, "off" to disable), never on
the public port. Besides the Go runtime and process metrics this adds:

- http_requests_total and http_request_duration_seconds per route pattern,
  from an Echo middleware.
- musicserver_songs_requested_total and musicserver_songs_played_total by
  pick mode (random, low, classic). The mode is now stored with each queued
  song (migration 000006, not released yet).
- Library size, sync duration, last successful sync, broken songs from the
  last full sync, and the database connection pool.

Refs #35, #36, #37

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

internal/backend/queue_integration_test.go

@@ -47,7 +47,31 @@ func TestQueueWithPostgres(t *testing.T) {
 		assert.NotEqual(t, uuid.Nil, song.SoundtrackUuid)
 	}
 
-	path, err := m.PreviousSong(ctx, queueID)
+	var modes []string
+	rows, err := database.Pool.Query(ctx, "SELECT mode FROM queue_song WHERE queue_uuid = $1 ORDER BY song_no", queueID)
+	require.NoError(t, err)
+	for rows.Next() {
+		var mode string
+		require.NoError(t, rows.Scan(&mode))
+		modes = append(modes, mode)
+	}
+	require.NoError(t, rows.Err())
+	assert.Equal(t, []string{"random", "random", "random"}, modes)
+
+	_, err = m.RandomSongClassic(ctx, queueID)
+	require.NoError(t, err)
+	require.NoError(t, m.AddLatestToQueue(ctx, queueID))
+	var classic string
+	require.NoError(t, database.Pool.QueryRow(ctx, "SELECT mode FROM queue_song WHERE queue_uuid = $1 AND song_no = 3", queueID).Scan(&classic))
+	assert.Equal(t, "classic", classic)
+	played, err = m.PlayedSongs(ctx, queueID)
+	require.NoError(t, err)
+	require.Len(t, played, 4)
+
+	path, err := m.Song(ctx, queueID, 2)
+	require.NoError(t, err)
+	assert.Equal(t, paths[2], path)
+	path, err = m.PreviousSong(ctx, queueID)
 	require.NoError(t, err)
 	assert.Equal(t, paths[1], path)
 	require.NoError(t, m.AddLatestPlayed(ctx, queueID))
@@ -82,7 +106,7 @@ func TestQueueWithPostgres(t *testing.T) {
 	require.NoError(t, q.ClearSongs(ctx))
 	played, err = m.PlayedSongs(ctx, queueID)
 	require.NoError(t, err)
-	require.Len(t, played, 3)
+	require.Len(t, played, 4)
 	assert.NotEmpty(t, played[0].Song)
 	_, err = m.Song(ctx, queueID, 0)
 	assert.ErrorIs(t, err, ErrNotFound)