Etale after base change If a morphism of schemes $f: X\rightarrow Y$ is etale after base change via an etale cover $Z\rightarrow Y$, is $f$ etale?

If $Z\rightarrow Y$ is etale and surjective, then your desired result follows from the fact that being etale is etale-local on the base. More generally, a morphism $f : X\rightarrow Y$ is etale if and only if there exists an fpqc morphism $Z\rightarrow Y$ such that the pullback of $f$ by $Z\rightarrow Y$ is etale.
Here, fpqc means faithfully flat and quasi-compact. In particular, fppf morphisms (faithfully flat and locally of finite presentation) are fpqc, and surjective etale morphisms are fppf.
In short, the property of a morphism being "etale" is fpqc-local on the base, hence a fortiori fppf-local on the base, and hence a fortiori etale-local on the base.
In fact, a whole slew of properties of morphisms are fpqc, fppf, or etale local on the base.
The chapter on descent:
http://stacks.math.columbia.edu/tag/0238
gives a long list of examples of such properties.
