Hey abscissa.

Do you mean finding the marginal distributions of X and Y given a joint distribution U = f(X,Y)? As you have pointed out since X and Y are independent then the joint distribution is P(X = x, Y = y) = P(X=x)*P(Y=y) and integrating out one will give the other.

Use the theorem to show that the sum of normals has the means summed and the variances summed as well.