Suppose that, under the conditions of CLT, $X_n\rightarrow N(\mu,\sigma^2)$ in distribution, what can we say about mean and variance of $X_n$?
$\mathbb{E}\left[X_n\right]\rightarrow\mu$? Or maybe$\,\,\,\,\,\,\,\mathbb{E}\left[X_n\right]\rightarrow\mu$ in distribution?
$Var\left[ X_n\right]\rightarrow \sigma^2$? Or $Var\left[ X_n\right]\rightarrow \sigma^2$ in distribution?
No. Here is a counterexample. Let $Z\sim N(0,1)$ and let $X_n=Z$ with probability $1-1/n$ and equal to $Z+n!$ with probability $1/n$. Then $X_n$ converges in distribution to $N(0,1)$ but the moments of $X_n$ do not behave nicely at all.
As @Shashi pointed out, the convergence in distribution of $EX_n$ is a mistake. $E X_n$ is not a random variable, so the best interpretation of "$EX_n$ converges in distribution" is that the sequence of numbers $EX_n$ converges. Which, as we have seen, does not necessarily happen.