I have some questions such that the corresponding statements are well-known for affine varieties, and I wonder whether they hold for projective ones.

Let $Z\subset X$ be a closed subvariety of a (projective) variety over a field $K$. Let $L/K$ be a finite field extension, and let $Y/X_L$ be an etale neighbourhood of $Z_L$ in $X_L$ i.e. $Y/X_L$ is etale and $Y\times_{X_L}Z_L=Z_L$. Is it true that $Y$ descends to an etale neighbourhood $U$ of $Z$ in $X$ i.e. that there exists an (etale neighbourhood) $U$ such that the morphism $U_L\to X_L$ factorizes through $Y$? Would it help if I will demand that $L/K$ is separable or Galois?

A reference question. For a domain $R$ and an extension $L$ of the fraction field of $R$ one can consider the integral closure (or the normalization) of $R$ in $L$. Now, Theorem 5.1 here http://mathsci.kaist.ac.kr/~jinhyun/note/normalization/normalization.pdf yields that a similar fact holds for any (irreducible) variety $V$ (instead of the spectrum of $R$) and a finite extension of the function field of $V$. Is there a 'canonical' reference for this fact? How would you call the variety obtained?

The integral closure opeation for (commutative) rings commutes with etale base change by [EGAIV, Prop. 18.12.15]. Does this statement generalize to varieties?