bittrex is a very good option, but I'm not sure if it has iota trading options. Binance is actually very good, you can try it, because so far, Binance has not been any bad news broke, it is relatively safe.
Why would you store any crypto in any exchange? the whole point of crypto is to hold your own private keys so they can never be hacked (assuming that your computer is not compromised already, or that you put it cold storage).
Then again, this IOTA new is not even a blockchain, and im not even sure if they have a wallet. I assume that they do... if this is thecase, then download it and put the private keys, or whatever they call it, on it, this way you will be safe, there's no other way. Again: never trust exchanges.