Does the extension reward me by mining in the background? (No!)

No! AirTime is not a browser miner. 

The AirTime payments are distributed from a pool of coins taken from the total block reward of the blockchain. The mining of the blockchain is performed by professional miners, not you as an AirTime user. A part of the block reward that miners generate (45%) is allocated to AirTime payments, which in turn are proportional based on global AirTime traffic.