The étale fundamental group as a functor

algebraic-geometry group-theory category-theory etale-cohomology

I hope you don't mind if I post an answer to get this question off the unanswered list. The answer to the first question is essentially given by Zhen Lin in the comments:

You should be able to find in standard references that the class of finite (resp. étale, surjective) morphisms is closed under pullback. For instance, try the Stacks project. – Zhen Lin Jan 25 '16 at 19:19

And the answer to the second question is at the end of the question statement.

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