The song list gets updated from time to time. That’s the only thing that I’m aware has changed. I think you might have just gotten lucky with songs being included in the list before and now, possibly after an update to the list, are less lucky.
AFAIK it only shows the song immediately if the phone can identify it offline. It can only do that with a limited set of songs so if you listen to something less known it won’t be able to recognise it without talking to google first. When you click the music symbol on the lock screen it actually uses online services to identify what’s playing.
Yes, because that’s using Google’s servers to identify the music. It’s not offline anymore which is why it’s not doing it by default.