Property of eigenvalues of gaussian random matrix of form $MEM^T$, where $E$ is a PSD diagonal matrix

Question

I have a gaussian random matrix $M_{n\times n}$, where $M_{ij}\sim N(0,1)$ i.i.d., and I have a given positive semi-definite diagonal matrix $E$, where $E_{ii}=\lambda_i\geq0$, and $E_{ij}=0\quad\forall \ i\neq j$.

I wonder can I state some distributional property of eigenvalues of $Q=MEM^T$? I suspect Marchenko-Pastur Law might be useful but I am not sure.

Can I claim that the rank of $Q$ is $n$ with high probability, where $n$ is the number of non-zero $\lambda_i$?