Do I need (should) keep the cryptos in other wallet like MTW or MyCrypto? Will my balance on exchange become zero if I move? I am not sure about the procedure of that.
Yes absolutely! You should always keep your coins on a wallet, of which you (as the only one) control the private keys. It doesn't matter which Ethereum wallet software it is, there are several good options, and I would recommend Metamask. In terms of manageability, it is the simplest.