If I have two random variables, $X$ and $Y$, such that $f_{X,Y}(x,y)$ is jointly Gaussian, is it ALWAYS true that $f_{X}(x)$ and $f_{Y}(y)$ are Gaussian as well?
If $f_{X,Y}(x,y) \sim N(\mu_{XY}, \sigma_{XY}^{2})$ with correlation $\rho$, then $f_{X}(x) \sim N(\mu_{X}, \sigma_{X}^{2})$ and $f_{Y}(y) \sim N(\mu_{y}, \sigma_{y}^{2})$.
Is there an exception?
Yes, it is always true. If $X$ and $Y$ are jointly Gaussian with mean $\mu = (\mu_x, \mu_y)$ and covariance matrix $\Sigma$, then $X$ is Gaussian with mean $\mu$ and variance $\Sigma_{1,1}$. Just integrate out the other variable: $$f_X(x) = \int_{-\infty}^\infty f_{X,Y}(x,y)\; dy $$