But isn't there some link between ICO and traditional cryptocurrencies? I thought the transactions are somehow logged to e.g. ethereum blockchain?

Most tokens are implemented by ethereum contracts, because it is the simplest way to implement such thing, but they don't have to be.

Here is where I go confused. New tokens are supposed to be completely independent from ethereum economically, but at the same time they are somehow ethereum contracts. Would that not mean that someone needs at some point own ethereum and take the market risk?

Yes, to transact tokens that run on top of Ethereum you have to pay transaction fees in ETH, which AFAIK means you need to buy at least a small amount of ETH. There are some recent proposals that would allow fees to be paid in other tokens, so I guess if those go through then you could use tokens without ever touching ETH.