Law of Iterated Expectations Explanation I am having trouble following a short derivation that uses the Law of Iterated Expectations that is found in the answer to another question: How to derive a regression formula
I will repeat it here:
Let $E(y|z) = \mu_{y|z}.$   Then it is shown that $E(y \mu_{y|z}) = Var(\mu_{y|z})$ in the following steps:
(1)  $E(y \mu_{y|z}) = E(E(y|z, \mu_{y|z}) *\mu_{y|z})$
(2)  $~~~~~~~~~~~~~= E(E(y|z) *\mu_{y|z})$
(3)  $~~~~~~~~~~~~~= E(\mu^2_{y|z})$
(4)  $~~~~~~~~~~~~~= Var(\mu^2_{y|z})$
I don't know all the properties of the LIE, but I do know that in general it gives $E(W) = E_Z(E(W|Z))$.  With that said: 
Q1. In line 1, what is going on?  How is the LIE being applied in this way?
Q2. In going from line 1 to line 2, why do we not condition on $\mu_{y|z}$ anymore? 
Q3. In going from line 3 to line 4, why is $(E(\mu_{y|z}))^2 = 0$ so that we get the variance?
 A: By the defining property of the conditional expectation,
$$E\big[E(W\mid Q)\big] = E(W)$$
Set $W \equiv y\cdot E(y\mid z)$ and $Q \equiv z$. Substitute to get
$$E\Big(E\big [y\cdot E(y\mid z) \Big|z\big]\Big)= E\big [y\cdot E(y\mid z)\big] $$
The right-hand-side is what you are starting with, so we need to manipulate the left-hand-side.  
By the measurability of $E(y\mid z)$ with respect to $z$ ("take out what is known") we have
$$E\Big(E\big [y\cdot E(y\mid z) \Big|z\big]\Big) = E\Big(E(y\mid z)\cdot E\big [y \mid z\big]\Big)$$
$$=E\left(\left[E(y\mid z)\right]^2\right) = {\rm Var}[E(y\mid z)] +\big(E\left[E(y\mid z)\right]\big)^2$$
$$={\rm Var}[E(y\mid z)] +\big(E[y]\big)^2$$
So for the stated result to go through, it must be the case that the unconditional expected value of $y$ is zero. This is of course not a general result, but it is an assumption explicitly made in Angrist's paper (which is used in the CV thread the OP linked to), where just above his eq. $(2)$ the author writes:   

"Let $\beta$ denote the population regression coefficient from a
  regression of (mean zero) $y$ on $μ_{y|z} = E[y|z]$, for any random
  variables, $y$ and $z$."

