Answering Frobenius norm part of the question:
Suppose $X$ contains $b$ IID instances of random variable $x$ stacked as rows. Let $x$ be distributed as zero-centered Gaussian with covariance $\Sigma$. We can show the following
$$E[\|X'X\|_F^2]=b(b+1)\operatorname{Tr}\Sigma^2+b (\operatorname{Tr} \Sigma)^2$$
To prove this, first note that:
$$\|X^T X\|_F^2=\operatorname{Tr} X^T X X^T X$$
And that for arbitrary R.V. $x$ we have $$E[X'XX'X]=bE[xx'xx']+b(b-1)E[xx']E[xx']$$
For Gaussian $x$ centered at zero we can apply Wick's theorem to get:
$$E[xx'xx']=2E[xx']E[xx']+E[xx'] \operatorname{Tr}E[xx']$$
Combining these three equations yields the result above