Where does the BTC come from when I buy through an Exchange?

Clarification would be appreciated!

When a Bitcoin miner mines BTC, I understand that it's adding more BTC into circulation, but is the BTC that is added into circulation the actual reward they're receiving for successfully mining? Then, the miner could do a few things with it like:
- sell their BTC that they received through an exchange
- transfer it to someone
- keep the reward

Is this correct?

Also, do exchanges have to first "buy" or "mine" BTC in order to sell it to the general public? If this is true, if the exchange has no BTC, is it possible for a customer to still buy BTC through that exchange?

Thanks!