pm97 Does Play services require unnecessary and invasive permissions to function correctly?
Yes, which is why GrapheneOS has GmsCompat, the compatibility layer that allows us to use Google Play as a regular non-privileged app.
pm97 Does Play services have the space to function correctly regarding notifications within the Graphene environment?
I'm not sure I follow what you mean here, but Google Play works perfectly for most people. Sure, there are some issues that pop up from time to time, but the developers are good about fixing those problems quickly.
pm97 Could it be the app that is failing, or does the notification sound first and then the app fails (if the app is failing)?
I suppose there has to be an explanation for what you're experiencing, but I'm not sure it would be this. Google Play may crash and you'll be able to see the crash log. Without seeing popups saying it crashed, I wouldn't assume that's what's happening.