If noise is normal does that mean endogenous variable is also normal? Say I have a regression model
$Y = XB + e$
Then if $e$ is Gaussian White Noise with $~N(\mu, \sigma)$
Does that mean Y is also normal?
Just checking my understanding.

NO
For an easy counterexample, let your lone feature be a binary indicator variable, and let the coefficient be nonzero. Then you have a bimodal marginal distribution of $Y$ (assuming the feature has indicators for both classes).
If you want to say that the marginal distribution of $Y$ is Gaussian under a null hypothesis that all (“all“ is important) of the features have corresponding parameters equal to zero, that would be reasonable, but then the null hypothesis is just $y_i=\beta_0+\epsilon_i$, so you just shift the $\epsilon_i$ distribution up or down the real line to have a mean of $\beta_0$.
