The key statement to verify is whether the thing you're looking at commutes with localization. If it does, then it commutes with sections on affine opens: start by defining the presheaf which takes the naive value on every open set, and then note that on principal affine opens $D(f)$ you get exactly the value you would expect from the quasi-coherent sheaf associated to the operation on the module of global sections. As the principal affine opens form a basis for the topology on an affine scheme and a sheaf can be specified by its values on a basis, the result follows.

In particular, this means that kernels, cokernels, images, and any other "homological" constructions commute with the associate sheaf functor on an affine scheme because localization is exact. Tensor products and nilradicals commute with localization, so they commute with the associated sheaf functor, too. Annihilators don't always commute with localization, so you need to be more careful with them.

Let me just mention one alternate neat proof that tensor products commute with sheafification, which I originally saw at the Stacks Project: consider the morphism of ringed spaces $\pi:(\operatorname{Spec} R,\mathcal{O}_{\operatorname{Spec} R})\to (\{pt\},R)$. Then the associated sheaf functor is $\pi^*$, which means it commutes with tensor products by the standard proof.