Yes of course you can transfer from exchanger to your personnal wallet but the fee will be applicble.
Make sure you know the time of transaction like recently the transction is very slow
Correction; The fees are extremely expensive, and transactions are slow.
So no, if you want to avoid transaction fees, you are not going to find anything that suits to you, because all the wallets, doesn't matter if it is a segwit one, or whatever it is , will let you send TX's with less than $2 worth of fees.
Good luck with that.