Finiteness of surjective etale morphisms Is every surjective etale morphism from a connected separated scheme to $A^n_{\mathbb{C}}$ of finite type? Is it finite? We use Stacks project's definitions. 
EDIT: From Jason Starr's answer, we learn that such a morphism indeed has to be of finite 
type, and since etale morphisms are locally quasi-finite, we infer that the morphism has to be quasi-finite. 
Is it true that every surjective etale morphism from a connected separated scheme to $A^n_{\mathbb{C}}$ such that the cardinality of the fiber over a closed point is independent of the choice of the closed point is finite? I think that for $n=1$, this question should be answered positively by considering local affine coordinates for etale morphism and applying the fact that a univariate complex polynomial has a non-simple root iff its derivative has a common root with it. I am not sure about $n>1$ though. 
 A: The question is "really" about quasi-compactness, which is usually assumed as a hypothesis in versions of Zariski's Main Theorem.  However, the other strong hypotheses of the OP imply quasi-compactness in this case.  The key point is that an open immersion is quasi-compact if the target is Noetherian.
Lemma.  Let $i:X\to Z$ be a separated morphism between irreducible schemes.  If there exists a covering of $X$ by open affines $U$ such that each restriction $i|_U$ is an open immersion, then $i$ is an open immersion.  If $Z$ is Noetherian, then $X$ is quasi-compact.
Proof.  Up to replacing $Z$ by the open image of $i$, assume that $i$ is surjective.  The goal is to prove that $i$ is an isomorphism.  We construct the inverse isomorphism $i^{-1}:Z\to X$ by gluing.  Let $U$ and $V$ be nonempty open subschemes of $X$.  The cocycle condition for $i^{-1}$ is precisely the condition that $i^{-1}(i(U)\cap i(V))$ equals $U\cap V$.
Let $Y^o$ be a nonempty open affine subset of the open intersection $i(U)\cap i(V)$.  Denote by $X^o$ the inverse image $i^{-1}(Y^o)$. Since $X$ is irreducible, the intersections of nonempty open subsets $U\cap  X^o$ and $V^\cap X^o$ are dense. Denote these by $U^o$ and $V^o$.  By construction, each of the following restrictions of $i$ is an isomorphism, $$i_U:U^o\to Y^o, \ \ i_V:V^o\to Y^o.$$  These isomorphisms agree on $U^o\cap V^o = (U\cap V)\cap X^o$.  
Since $i$ is separated and since $Y^o$ is affine, the scheme $X^o$ is separated.  Define $j$ to be the automorphism of $X^o$ whose restriction to $U^o$ equals $i_V^{-1}\circ i_U$ and whose restriction to $V^o$ equals $i_U^{-1}\circ i_V$.  These glue since $i_U$ and $i_V$ agree on $U^o\circ V^o$.  Moreover, $j$ equals the identity on $U^o\circ V^o$.  Since $j$ and the identity agree on the dense open $U^o\circ V^o$, and since $X^o$ is separated, the morphism $j$ equals the identity.  Thus, $U^o$ equals $V^o$.  Since we can cover $i(U)\cap i(V)$ by such open affines, it follows that $i^{-1}(i(U)\cap i(V))$ equals $U\cap V$.  
Finally, if $Y$ is Noetherian, then every open subset of $Y$ is quasi-compact.  Thus, the scheme $X$ is quasi-compact. QED
Let $f:X\to Y$ be a locally finite type, separated morphism with finite fibers that is quasi-finite Zariski locally on $X$, and that is strongly dominant in the sense that the $f$-inverse image of every dense open subset of $Y$ is a dense open subset of $X$. Assume also that $X$ is normal and that $Y$ is quasi-compact, separated, excellent, integral, and normal.  
Proposition.(Variant of Grothendieck's "Zariski Main Theorem") There exists a factorization of $f$ as the composition of a dense open immersion into a normal scheme, $i:X\hookrightarrow Z$, followed by a finite, strongly dominant morphism, $g:Z\to Y$.  Moreover, this factorization is unique up to unique isomorphism.
Proof.  Every irreducible component of $X$ dominates $Y$, i.e., every generic point of $X$ maps to the generic point of $Y$.  By hypothesis, there are only finitely many preimages of the generic point of $Y$, i.e., $X$ has only finitely many irreducible components.  Since $X$ is normal, these irreducible components are connected components.  Without loss of generality, assume that $X$ is connected, i.e., $X$ has a unique generic point $\eta$. 
Since $Y$ is excellent, the "integral closure" of $Y$ in the "function field" $\kappa(\eta)$ is a finite, strongly dominant morphism whose domain is normal, $$g:Z\to Y.$$  By the universal property of the normalization, there exists a unique morphism of schemes compatible with the specified morphisms to $Y$, $$i:X\to Z.$$  By Zariski's Main Theorem, working locally on $X$ with opens that are quasi-compact over $Y$, the morphism $i$ is an locally an open immersion. 
By the lemma, the morphism $i$ is an open immersion.QED
Now you can apply this when $Y$ is affine space.  I recommend that you read Grothendieck's formulation of Zariski's Main Theorem in EGA.
