Is Firebase Cloud Messaging free?

There is no such thing "unlimited" for free. If you will send billions of billions messages a day, you will likely hit the maximum throughput limits and get blocked, or perhaps some sales man will contact you.

Firebase Cloud Messaging is free of charge, just like Google Cloud Messaging was. The Firebase pricing page shows exactly what Firebase features are paid.

As of April 10, 2018, Google has deprecated GCM. The GCM server and client APIs are deprecated and will be removed as soon as April 11, 2019. Migrate GCM apps to Firebase Cloud Messaging (FCM), which inherits the reliable and scalable GCM infrastructure.

Yes, it's free. You can use Analytics for advanced messaging - out of the box.

Which products are paid? Which are free?

Firebase's paid infrastructure products are the Realtime Database, Firebase Storage, Hosting, and Test Lab. We offer a free tier for all of these products except Test Lab.

Firebase also has many free products: Analytics, Notifications, Crash Reporting, Authentication, Remote Config, Dynamic Links, Firebase Invites, and App Indexing. You can use an unlimited amount of these in all plans, including our free Spark Plan.