The device supports instant playback and the playback should start instantly with no delay.
Could you please make sure the Menu/recording/pre-recording turned off on your device?
Does the song has midi too?
If not could you please simply drag and drop the first wav file from the song folder on the USB drive to any DAW and check the song start with no silence on the beginning?
Unfortunately to start another song in the same time is not possible, however the device supports gapless playback and you can change the next song during the playback too.