What is the Sun's core made of?
The core fo the sun is make of energy. The core of the sun is composed of a soup of protons, neutrons, and electrons. The temperature is so high that these cannot possibly unite into elemental atoms. The pressure is so great that the particles are compressed into a dense mass, and and the heat and pressure allows Fusion to take place, making heavier proto-elements out of the simple feedstock. In addition, there is carbon, iron, silicon, oxygen, sulfur, and other elements with an atomic weight less than iron. You must understand that these "elements" do not exist in the form that we regularly encounter on Earth. They are the end products of atomic FUSION, and because of the incredibly high temperatures in the sun's core the electrons are stripped away and will not be reunited with the diverse atomic nuclei until either: 1.) they make their way to the surface and are ejected into space where the cooler temperatures allow the electrons to unite with the proto-elements, or 2.) the sun eventually balloons into a red giant star, fuses more heavy elements, collapses, and explodes as a nova.
I'm not sure I understand your question. Molecules are made up of atoms, elements are composed of the same type of atoms. Every type of naturally occurring atom (except hydrogen & helium) is formed in the core of Suns (stars) and no where else. Since molecules are made of atoms and atoms are made in Suns you could say all molecules are synthesized in Suns.
The Sun does not have a definitive surface like the inner planets, rather, the density of the gas increases as you go down towards the core. I presume you are referring to the photosphere when you say 'surface'. In this case it is made of what the rest of the star is made of: Primarily hydrogen with some helium.