Only notifications from text messages are being shown
It sounds like you've gone into the developer options and turned on "limit background processes", which stops apps running in the background to do all these things. The fact that they work normally when you open the app by hand suggests that app is still getting the information it needs, just not until it's in the foreground. The fact it's the whole behaviour of the app (e.g. Play Store downloading updates), not just the appearance of the notification tells me it's not just a problem with notifications.