Suppose $X \mid Y$ and $Y$ are normally distributed. Does it follow that $Y \mid X$ is normally distributed? Suppose $X \mid Y$ and $Y$ are normally distributed. Does it follow that $Y \mid X$ is normally distributed?
And if so, how would one prove this?

Here, as a counterexample, is a sample of points where $Y$ has a standard Normal distribution and the conditional distribution of $X$ is always Normal, too.

This is obviously non-Normal.
To guarantee Normality, you need (almost surely) that
(1) $E[X\mid Y]$ must be a linear function of $Y$ and
(2) $\operatorname{Var}(X\mid Y)$ must be constant.
These are both characteristics of any Bivariate Normal distribution, so they are necessary conditions.  When you write down the joint distribution implied by both these conditions, it will be Gaussian: that is, Bivariate Normal.
These R commands generated the example.  The conditional variance $Y^4$ is not constant.
n <- 1e3
y <- rnorm(n)
x <- rnorm(n, y, y^2)
plot(x,y, col = "#00000040") # Semi-transparent points


