Let $X\sim \mathcal N(\mu_X,\sigma_X^2),\ Y\sim \mathcal N(\mu_Y,\sigma_Y^2)$ two normal random variables and $a,b\in \mathbb R$.
If $X,Y$ are independent, then $$aX+bY\sim \mathcal N(a\mu_X+b\mu_Y,a^2\sigma_X^2+b^2\sigma_Y^2)$$ If $X,Y$ are jointly normally distributed with correlation $\text{corr}(X,Y)=\rho$, then $$aX+bY\sim \mathcal N(a\mu_X+b\mu_Y, a^2\sigma_X^2+b^2\sigma_Y^2 + 2ab\rho\sigma_X\sigma_Y)$$
But what if $X,Y$ are neither independent nor jointly normally distributed, what can we say about the distribution of their sum? And what would be an example of a sum of normal random variables that is not normally distributed?
Actually, your second statement is false. This would only be true if $X,Y$ are jointly normal, which you do not assume. And if you're assuming joint normality then uncorrelated=independent.