An exchange is where you buy/sell cryptocurrencies. A wallet is where you keep cryptocurrencies you’ve bought or have received.

No, an exchange cannot be a wallet. However exchanges usually come with a wallet.

Coinbase, for instance, is an exchange and the account you create with them already comes with a wallet from them. This wallet is controlled by you but “owned” by them. I say owned because if they’re website crashes you have no access to “your wallet”. That’s why most people suggest you create a wallet using example MyEtherWallet where you are in complete control.

No. You need an exchange or interface to send or purchase cryptocurrencies. However you don’t need any of these to receive cryptocurrencies from someone who is sending to your wallet address. You don’t even need to be online. It’s rather that person who needs the interface to send to you.