It is known, that the étale fundamental group of a normal connected scheme equals the galois group of the maximal unramified extension of its function field.

This is not true for integral schemes in general. The point is somehow, that one cannot construct flat finite covers from a function field extension as one does with normalization. So my questions are as follows:

What are examples for integral schemes, where the fundamental group is not a galois group of the function field? Or equivalently: A scheme with non-isomorphic étale covers but isomorphic function fields?

Is there already an example which is one-dimensional, e.g. an algebraic curve or an order in a number field? Somehow the étale covers should look different over the singularities. What can happen there?

In the case of curves it is known how to construct singular curves from normal ones. Is there a way of constructing étale covers of a singular curve by taking its normalization in a function field extension and then singularizing it again?

I thank you very much for your answers.

connectedfinite etale cover of a non-normal irreducible (noetherian) $X$ is determined up to isomorphism by its generic fiber. In ACL's answer, the connected finite etale covers are determined up to isomorphism by their generic fiber. – user29283 May 5 '13 at 15:18