Right now you're only buying ERC20 tokens. Once the EOS platform is launched and people receive their 1:1 tokens on the EOS blockchain, then the IQ tokens will be distributed on the EOS blockchain.

Yes you need to register your MEW wallet to an EOS wallet before Jun 1. Once the wallet is registered just transfer all your EOS tokens to it (The MEW Wallet). You only need to register the wallet once so if you want to buy more on exchanges and send them to the same wallet later, you dont have to re-register them - the wallets are effectively tied so that when the Jun 1 snapshot is taken, your EOS wallet (tied to the MEW wallet) will have equal tokens. Also any airdrops like IQ will be credited.

what happens if I do not claim my token when public sale is end? I still cannot figure out the way to claim tokens

If you hold EOS Tokens in an Ethereum account and fail to register your Ethereum address to your EOS public key, then your EOS Tokens may not be part of a Snapshot (as defined below) a third party may wish to create when the EOS Tokens become fixed and non-transferable on the Ethereum blockchain. In order to be part of such Snapshot, it is important that you generate an EOS public and private key pair (see the “Generate EOS keys” button below) and register your EOS public key with your Ethereum address prior to the EOS Tokens becoming fixed and non-transferable on the Ethereum blockchain which will occur within 23 hours after the end of the final period on June 1, 2018 at 22:59:59 UTC.

Currently i can store my EOS using a hardware wallet, once the EOS blockchain gets released in June how will we be able to store it? Will any hardware wallets have EOs?