Signal developer here. Not entirely sure what you're saying. I'm only an Android guy, but FCM messages are certainly one trigger that can allow an app process to run, but it's not the only trigger. You can schedule system alarms, jobs, etc. And the notification does not need to be provided by the FCM message. In our case, the server just sends empty FCM messages to wake up the app, we fetch the messages ourselves from the server, decrypt them, and build the notification ourselves. No data, encrypted or otherwise, is ever put into the FCM payloads.
Sure but it needs to go through Firebase regardless of the content of the notification message, I do not believe there is a way to use a third party notification service which does not depend on Firebase.