(1) If I have $X,Y$ noetherian schemes, locally of finite type over a field $k$, normal and connected, can I somehow conclude that a surjective étale morphism $X \to Y$ is finite? I have seen some counter examples to similar situations, but none of them in these full hypotheses.
(2) The more specific question that I am interested in is whether in the situation above $X \times_Y X$ is a disjoint union of copies of $X$ whose open embeddings compose to the identity with the projections.
Thanks in advance for any help.
Unfortunately no. Let $f\colon X'\to Y$ be a connected finite etale cover of degree $2$ (let's assume $k$ is algebraically closed for simplicity) where $Y$ is normal and connected. Then, consider $f|_{X}$ where for some point $p\in Y(k)$ with $f^{-1}(p)=\{q_1,q_2\}$ we set $X=X'-\{q_1\}$. Then, evidently $f|_{X}$ is etale and surjective, both normal connected, but is certainly not finite (e.g. use the fact that fiber size over $k$-points is constant for finite etale morphisms with target connected).