Mean independence and normality imply independence?

Question

Let $\varepsilon, X$ be real-valued random variables. Could you help me to show (if correct) that

(1) $E(\varepsilon)=E(\varepsilon\mid X)$

and

(2) $\varepsilon \sim N(0,1)$

imply that $\varepsilon\perp X$?

If the statement is wrong what would be the "corrected" version?

Answer 1

Suppose $X=\epsilon^2$, and $\epsilon\sim N(0,1)$.