You are correct that there are some errors in the Push Notification demo project due to compatibility issues. Hopefully this is corrected in the AppStore version soon. To get it to work try the following:
1) Start a new Mendix 7 project
2) Download the Mendix Push Notification module
3) For all the errors about navigation layouts.sidebar_full_responsive, pick a new layout. Atlas_topbar is the new default so you can pick that one to fix the errors. I had to change this twice to fix all wrrors
4) Download the required modules: encryption, community commons
Now you have what is effectively the demo project for the Push Notification set up for Mendix 7, however there is a known bug with the module that is being worked through on the GitHub. There is a newer version of the module that is still being tested and not yet in the app store, but you can see all the information about this and get a fix:
So now that you have the technical pieces in place, how can you actually do anything with it? Well from there you can follow the instruction guide, which you have likely already seen:
This guide does include a section about setting up Firebase Cloud Messenger specifically, which can also be found here:
Hopefully this gives you enough to get started. Once you add the push notification administration snippit to your project, you can follow this guide on how to send a test push notification:
Then you can look at the microflow that actually sends the push notification (in a Java action) and use this for your own notifications.
I hope this is helpful to you.