I use browser facebook on phone and it gives me notifications.
It's basically when I deactivated that that I thought "I wouldn't mind 22 notifications from BM, but facebook, come on..."
I think email for biggish things would be good.
They can both be opt-in but make it as low-key or high-key as the user wants?
Also, I'd really like to get an email a day or two before a character autopauses... and a warning on my family screen (though this is a different feature request).