Skjold Yes, as has been said, Proton's apps work exactly as they should with no Play Services on the phone. They can also be downloaded directly from Proton, and updated using Obtanium so no Play Store account is required.
Sadly, due to a lack if technical knowledge, I am unable to say exactly what Firebase does, except to say that it sends/receives pings to your phone from Google regarding notifications. It is also doing this regardless of whether notifications are turned on or off in the Proton mail app. This was enough for me to uninstall the app and revert to using my browser for Proton mail.
Since then, I have created another profile just for the mail app which I can switch to occasionally for convenience, but is inactive at all other times.