So do the coins, such as the ICO's created on the Ethereum network, operate on Ethereum's blockchain, such as within a sidechain I guess, or will they also create their own singular blockchain for their coin once it is established? Thank you in advance.
Ethereum tokens don't have their own blockchain, they operate and rely completely on Ethereum blockchain. These tokens are just smart contracts running on ETH blockchain. You will need to pay the tx fees/gas in ETH to send these tokens and the fee/gas will go to the ETH miner.