In linear regression, are the error and the predictor variable independent? We have a simple linear regression model. Our assumptions are:
$Y_i =\beta_0+\beta_1X_i+ \varepsilon_i $, $i=1, \cdots, n$
$\varepsilon_i \sim N(0, \sigma^2)$
$Var(\varepsilon_i|X_i=x)=\sigma^2$ 
$\varepsilon_1, \cdots, \varepsilon_n$ are mutually independent.
$\\$
Are these hypothesis enough to claim that $\varepsilon_i|X_i=x \sim N(0, \sigma^2)$?
 A: The assumption that the conditional variance is equal to the unconditional variance, together with the assumption that $E(\varepsilon_i)=0$, does imply zero conditional mean, namely
$$\{{\rm Var}(\varepsilon_i \mid X_i) = {\rm Var}(\varepsilon_i)\} \;\text {and}\;\{E(\varepsilon_i)=0\}\implies E(\varepsilon_i \mid X_i)=0 \tag{1}$$
The two assumptions imply that 
$$E(\varepsilon_i^2 \mid X_i) -[E(\varepsilon_i \mid X_i]^2 = E(\varepsilon_i^2)$$
$$\implies E(\varepsilon_i^2 \mid X_i) - E(\varepsilon_i^2) = [E(\varepsilon_i \mid X_i]^2$$
Ad absurdum, assume that $E(\varepsilon_i \mid X_i)\neq 0 \implies [E(\varepsilon_i \mid X_i]^2 >0$
This in turn implies that  $E(\varepsilon_i^2 \mid X_i) > E(\varepsilon_i^2)$. By the law of iterated expectations we have $E(\varepsilon_i^2) = E\big[ E(\varepsilon_i^2 \mid X_i)\big]$. For clarity set $Z \equiv E(\varepsilon_i^2 \mid X_i)$. Then we have that
$$E(\varepsilon_i \mid X_i)\neq 0  \implies Z > E(Z)$$
But this cannot be since a random variable cannot be strictly greater than its own expected value. So $(1)$ must hold.  
Note that the reverse is not necessarily true.
As for providing an example to show that even if the above results hold, and even under the marginal normality assumption, the conditional distribution is not necessarily identical to the marginal (which would establish independence), whuber beat me to it. 
A: No.  Here's an interesting counterexample.
Define a density function
$$g(x) = \frac{2}{\sqrt{2\pi}}\exp(-x^2/2)I(-t \le x \le 0 \text{ or } t \le x)$$
for $t = \sqrt{2\log(2)} \approx 1.17741$.  ($I$ is the indicator function.)
The plot of $g$ is shown here in blue.  If we define $h(x) = g(-x)$, its plot appears in red.

Direct calculation shows that any variable $Y$ with density $g$ has zero mean and unit variance.  By construction, an equal mixture of $Y$ with $-Y$ (whose PDF is $h$) has a density function proportional to $\exp(-x^2/2)$: that is, it is standard Normal (with zero mean and unit variance).
Let $X_i$ have a Bernoulli$(1/2)$ distribution.  Suppose $\varepsilon_i|X=0$ has density $g$ and $\varepsilon_i|X=1$ has density $h$, with all the $(X_i, \varepsilon_i)$ independent. The assumption about $Y_i$ is irrelevant (or true by definition of $Y_i$) and all the other assumptions hold by construction, yet none of the conditional distributions $\varepsilon_i | X_i = x$ are Normal for any value of $x$.

These plots show a dataset of $300$ samples from a bivariate distribution where $E[Y|X]=5 + X.$  The $x$ values in the scatterplot at the left have been horizontally jittered (displaced randomly) to resolve overlaps.  The dotted red line is the least squares fit to these data.  The three histograms show the conditional residuals--which are expected to follow $g$ and $h$ closely--and then the combined residuals, which are expected to be approximately Normal.
