Can the Google Play Store inside the private space install and update apps outside of it?
No.
Like how the Google Play Store on the owner profile can push apps to other profiles and update them, all from that owner profile.
That's not true. Sandboxed Google Play is a regular sandboxed app and cannot do things across profiles. It has nothing to do with how it works on the stock OS where Google Play are privileged OS components with privileged permissions to interact and operate across profiles among many other things. That is certainly not the case on GrapheneOS.
Packages installed across multiple profiles are shared and updating them in one will update them across the others. That's not specific to any particular installer.