Does $X-\operatorname{E}(X \mid Z)$ vary more or less then $X$? 
Does $X-\operatorname{E}(X \mid Z)$ vary more or less then $X$?

I've tried the following:
$$\begin{align}
\operatorname{Var}(\color{green}{X-\operatorname{E}(X \mid Z)})
&= \operatorname{E}\left[\operatorname{Var}(\color{green}{X-\operatorname{E}(X \mid Z)} \mid Z)\right] + \operatorname{Var}\left[\operatorname{E}(\color{green}{X-\operatorname{E}(X\mid Z)}\mid Z)\right]\\
&= \operatorname{E}\left[\operatorname{Var}(X\mid Z)\right] + \operatorname{Var}\left[\operatorname{E}(X\mid Z)-\operatorname{E}\left\{\operatorname{E}(X\mid Z)\mid Z\right\}\right]\\
&= \operatorname{E}\left[\operatorname{Var}(X \mid Z)\right]\end{align}$$
I don't see how I can go on from here.

By Cauchy-Schwarz inequality, $$E(E(X\mid Z)^2)\geqslant E(E(X\mid Z))^2=E(X)^2,$$ hence $$E(X^2)-E(E(X\mid Z)^2)\leqslant E(X^2)-E(X)^2,$$ that is, $$\mathrm{Var}(X-E(X\mid Z))\leqslant \mathrm{Var}(X).$$
