Consequences of expected value - true-false question

Question

True or false, prove or give a counterexample

If $Y = \alpha +\beta X+\epsilon$:

1. and $E[\epsilon\mid X] = 0$ then $E[X^2\epsilon] = 0 $

2. and $E[X\epsilon] = 0$ then $E[X^2\epsilon] = 0$

3. and $E[\epsilon \mid X] = 0$ then $\epsilon \perp\!\!\!\perp X$

I have some problems regarding problem 3. Could someone verify?

1: True

Since $E[X^2\epsilon] = E(E[X^2\epsilon\mid X]) = E(X^2 E[\epsilon\mid X]) = 0$

2: False

Counterexample:

Let $X\stackrel{d}{=} N(0,\sigma^2)$ and $\epsilon$ be a constant then $E[X\epsilon] = 0$ but $E[X^2] = 1$ and $E[X^2\epsilon] = \epsilon$.

3: False

Would the following meet the problem statement?

Let $\epsilon\mid X \stackrel{d}{=} N(0, X^2)$, then $E[\epsilon\mid X] = 0$ but they are not independent.