Technically the problem term would be $E[Y^2|X]$
For simplicity, let us for a minute assume that $y = a + bx$
Is it mathematically correct to write: $E[Y^2|X] = E[(a+bx)^2]$, and if so why?
And how would this apply to the direct formula of variance? Can we formulate
$$V[Y|X] = E[(Y - E[Y|X])^2 \ | \ X] = E[((a+bx)-E[Y|X])^2 ]$$
Also does anyone have a link where I can read more about this?
A conditional expectation is a function of the conditioning variable. Here, that is $X$.
For any function, $g$, the conditional expectation $\mathsf E(g(X)\mid X)$ will equal $g(X)$.
[ When given $X$, then you should surely expect $g(X)$ to be $g(X)$. ]
So when $Y$ is $a+bX$, then: $$\mathsf E(Y^2\mid X)=(a+bX)^2$$
Likewise for the conditional variance.$$\begin{align}\mathsf{Var}(Y\mid X) &= \mathsf E((Y-\mathsf E(Y\mid X))^2\mid X) \\ &=\mathsf E(((a+bX)-\mathsf E(a+bX\mid X))^2\mid X)\\&=0\end{align}$$
Note: You should anticipate this result. Since for any given value of $X$, we have that $Y$ is invariably $a+bX$.