Yes, APNs (Apple Push Notification service) has to be allowed for notifications to come through from messaging apps and the network operator can’t tell whether it’s an allowed messaging app or any other kind of notification.

The scale of Apple’s notification service must be pretty large. Granted, most notifications don’t have strong SLA guarantees but I don’t remember it having any downtime either.

They don't really make guarantees about the reliability of push notifications (IIRC), so it's unlikely you'd see anything about downtime unless it was sustained for some time.