You submit the action you want to do, anywhere in the ethereum network, and if it's valid (e.g. if you're running an operation that destroys a kitty then you'd better include a digital signature showing that you own that kitty) then it will be accepted and published. You have to pay a "gas fee" which in theory a) should be pretty small and b) is what pays for the running of that ethereum network. In practice under the current implementation those fees got to be big enough that they kind of killed cryptokitties, but this might be fixable.

What do you mean? All of the data is on the chain. It's a global "third-party" that everyone is a part of.