I have added AdMob advertising in my Android app .
I connected to the AdMob site using this link:
https://firebase.google.com/docs/admob/android/quick-start
AdMob integration can happen without Firebase it is just a recommendation by Google to use firebase along with it to get better app analysis.
Firebase is mostly free except a few features like TestLab and some features have set quota that comes under free tier, one can definitely use firebase along with AdMob and yes for free features nothing needs to be paid and even no need to add a payment method unlike other platforms.
Yes, you can use AdMob standalone, without Firebase. But I recommend to implement AdMob using Firebase. It will save you from lot of setup efforts and is simple to integrate in app.
For official site:
Firebase is recommended, but optional.
The best way to use AdMob is with Firebase, but for those publishers who aren't ready to make the switch just yet, Google Mobile Ads will remain available as a standalone SDK. Android publishers can continue to import the play-services-ads Gradle artifact without using the Firebase plugin.