Can I mine cryptocurrencies with Golem? Or would it be far more efficient and cost effective to simply convert Golem (GNT) into the desired currency on an exchange.
Golem is not a miner. It is a distributed Supercomputer. You can share your CPU and GPU with requestors and get GNT in return. And then you can withdraw and exchange GNT to other cryptos on an exchange. So basically there is no option to "mine" other cryptos on Golem.
Where can i store my GNT- MyEtherWallet OR Mist? or should i use a hardware wallet like the Trezor for additional security.
Are you serious when you ask this question? GNT is an ERC-20 token. You can store it on any ERC-20 compatible wallet. But prefer to store where you have control over the private keys.