However, there are two other types of neutrinos, the Muon and Tau Neutrinos. Does the Sun produce them? If not, are they produced naturally anywhere in the universe?
The neutrino creating reactions (fusions, decays, and fissions) that take place in the core of the sun are all low enough in energy that they can only produce electron neutrinos.
However, neutrinos mix (because they propogate in mass states which do not correspond to the flavor states in which they are produced and detected), and because the energy of the neutrinos is low (a few MeV or less) the length scale for that mixing is much less than a the solar radius.
The result is that the neutrino flux emerging from the sun is fully mixed and any experimental sampling of the flvors (say by the SNO experiment) will report that only about one third of them are electron flavored.
There can also be blackbody (thermal) antineutrino and neutrinos of all flavors, which are emitted during type-II supernovae core collapse.
Here is a figure from H.-Thomas Janka, Neutrinos from type-II supernovae and the neutrino-driven supernova mechanism (reprint from: Conference Proceedings Vol. 40: "Frontier Objects in Astrophysics and Particle Physics").