As the title states, we are asked to find two random variables $X$ and $Y$ such that $X+Y$ is the standard normal, providing the PDF of $(X,Y)$ and $E[XY]$.
So far I have read that the sum of two independent normal variables is normal, so I could take say $X$ with mean $-1$ and variance $-2$, and $Y$ with mean $1$ and variance $3$, and arrive at the standard normal for $X+Y$. However the proof I read was very hard to understand. Could anyone explain this process?
Your idea of adding two independent normals works, but variances are always non-negative, so variance of $-2$ doesn't make sense. Instead you can make both of the variances $1/2.$ It's also a little strange to make one of them mean $-1$ and the other mean $1,$ although this would work. The natural choice would be to take both means to be zero.
As for the proof, it would be best for you to say what the proof was that you read and what parts you didn't understand. I could throw a proof at you, sure, but I have no way of knowing whether it would be helpful.