as for new coins, miners do technically create bitcoin! when a miner finds a new block he is awarded by newly created coins which is a fixed amount (currently 12.5BTC) in other words they "create" a new transaction known as coinbase transaction which pays them this newly generated coins. it is not out of thin air, it is the design of bitcoin to distribute the 21 million supply in this manner.
I don't believe that this is true. The 21 million "coins" were created when the genesis block was laid, and there is no possibility of any more being created. Miners perform a number od SHA256 calculations, and when they discover a result that satisfies the current requirements, they are able to create block which is added to the blockchain. They are rewarded for their work by being able to claim a few coins from the unallocated pool of Bitcoin ( and the transaction fees for the transactions included in their block). Were miners able to create new coins, then there would be no 21 million cap, and no halving of the reward in my opinion.