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/testutil/fakedb/fakedb.go

@@ -431,7 +431,9 @@ func (s *Store) EndActiveQueues(context.Context) error {
 }
 
 func (s *Store) SetQueueLastFetched(_ context.Context, arg repository.SetQueueLastFetchedParams) error {
-	return s.updateQueue(arg.Uuid, func(q *repository.Queue) { q.LastFetchedSongUuid = arg.LastFetchedSongUuid })
+	return s.updateQueue(arg.Uuid, func(q *repository.Queue) {
+		q.LastFetchedSongUuid, q.LastFetchedMode = arg.LastFetchedSongUuid, arg.LastFetchedMode
+	})
 }
 
 func (s *Store) SetQueueCurrentSong(_ context.Context, arg repository.SetQueueCurrentSongParams) error {
@@ -457,6 +459,10 @@ func (s *Store) AddLastFetchedToQueue(_ context.Context, id uuid.UUID) error {
 	if st := s.soundtrackIndex(song.SoundtrackID); st >= 0 {
 		soundtrackName = s.Soundtracks[st].SoundtrackName
 	}
+	mode := "unknown"
+	if queue.LastFetchedMode != nil {
+		mode = *queue.LastFetchedMode
+	}
 	songNo := int32(0)
 	for _, qs := range s.QueueSongs {
 		if qs.QueueUuid == id {
@@ -465,9 +471,10 @@ func (s *Store) AddLastFetchedToQueue(_ context.Context, id uuid.UUID) error {
 	}
 	s.QueueSongs = append(s.QueueSongs, repository.QueueSong{
 		QueueUuid: id, SongNo: songNo, SongUuid: song.Uuid, SoundtrackName: soundtrackName, SongName: song.SongName,
+		Mode: mode,
 	})
 	queue.CurrentSong = songNo
-	queue.LastFetchedSongUuid = uuid.Nil
+	queue.LastFetchedSongUuid, queue.LastFetchedMode = uuid.Nil, nil
 	return nil
 }
 
@@ -481,7 +488,9 @@ func (s *Store) ListQueueSongs(_ context.Context, id uuid.UUID) ([]repository.Li
 		if qs.QueueUuid != id {
 			continue
 		}
-		row := repository.ListQueueSongsRow{SongNo: qs.SongNo, SongUuid: qs.SongUuid, SoundtrackName: qs.SoundtrackName, SongName: qs.SongName}
+		row := repository.ListQueueSongsRow{
+			SongNo: qs.SongNo, SongUuid: qs.SongUuid, SoundtrackName: qs.SoundtrackName, SongName: qs.SongName, Mode: qs.Mode,
+		}
 		if i := slices.IndexFunc(s.Songs, func(song repository.Song) bool { return song.Uuid == qs.SongUuid }); i >= 0 {
 			song := s.Songs[i]
 			row.Path, row.TimesPlayed, row.SoundtrackUuid = &song.Path, &song.TimesPlayed, song.SoundtrackUuid
@@ -492,6 +501,26 @@ func (s *Store) ListQueueSongs(_ context.Context, id uuid.UUID) ([]repository.Li
 	return rows, nil
 }
 
+func (s *Store) CountLibrary(context.Context) (repository.CountLibraryRow, error) {
+	defer s.lock()()
+	if s.Err != nil {
+		return repository.CountLibraryRow{}, s.Err
+	}
+	var count repository.CountLibraryRow
+	for _, st := range s.Soundtracks {
+		if st.Deleted != nil {
+			continue
+		}
+		count.Soundtracks++
+		for _, song := range s.Songs {
+			if song.SoundtrackID == st.ID {
+				count.Songs++
+			}
+		}
+	}
+	return count, nil
+}
+
 // Statistics queries return empty results; tests that need data use their own fakes.
 
 func (s *Store) GetMostPlayedSoundtracksWithSongs(context.Context, int32) ([]repository.GetMostPlayedSoundtracksWithSongsRow, error) {