Will money still have any value in the fully automated world?
Your argument is built on this assumption:


  When there is no work that people can do, I imagine that there are no costs associated with it, and consequently, everything is free for everyone


That assumption is not true, and so your conclusion does not stand.

We usually classify production factors into labour, land, capital, enterprise. As you can see, labour is only one of the four factors. But all four factors incur costs: land still needs rent, enterprise still needs profit, capital still needs investment.

Another way of looking at it, is that even if the supply of labour is effectively unlimited (because so little of it is needed to produce anything), the supply of other things is still limited, i.e. there is still some scarcity. There is still a limited amount of iron in the world, or copper, or cabbages. And money is part of markets, which are the main mechanism for managing those limited supplies.