They get a cut of app store sales, and additional third parties pay them to display ads within mobile apps using admob.

Although it isn't hard to come up with other ways for them to extract more money out of android, I can't imagine a lack of willing buyers for all sorts of information they could harvest from the billion or so deployed devices.

They certainly do charge for Android. AOSP is free, but that only includes a small part of what is shipped on an Android Phone. If you want all the apps (Mail, Play Store, etc.) you have to pay.

Play store