Ermm...Google don't charge for Android. That's the point. The manufacturers only have to pay for patent licensing for all the stuff Google ripped-off from other companies.
No money is received directly by Google when you buy an Android handset.
Ermm... No. Google charge for access to certain apps. What now? Ah yes, access to apps such as Gmail, Maps, Playstore etc. So, in fact, you do pay for the Gmail app already. If that's not enough to sustain their business, then maybe they should charge more for it? It's their business model after all, and they already push you to google search (obviously), which does indeed serve you adverts.