The first claim can be wrong if the elliptic surface has singular fibers with multiple components. This increases $R^2$ at those points, giving it a punctual subsheaf, giving it more global sections. We can associate these with the classes of fibral divisors that have zero intersection with the class of a fiber, say.
Your second claim follows from the general fact that the lowest filtered piece of the Leray spectral sequence comes from functoriality of cohomology, so the classes in it are all pullbacks.
I don't know a reference.