"Is it possible to install the Google Play Store?

Yes, this is generally possible. However Google doesn't allow anyone to ship its applications as long as the device is not certified and the vendor didn't sign an agreement with Google.

The Anbox project does not have any interest in shipping the Google Play store and we're not allowed to do so. We may add an easy way for our users at a later point which allows easy distribution of Android applications suited for the Anbox runtime environment."

speaking of which, is there a generic push notification system available that doesn't require Google Play Services?

doesn't seem that hard in theory. websocket service that instructs the creation of an android notification object