Since the neutrino flies through everything at the speed of light, it appears to have no mass. Is it just pure energy? Is E=mc² correct if m=0?

Answer

A neutrino does not fly through everything, it can very, very exceptionally interact with matter. A gigantic neutrino detector of 1 cubic kilometer has been built at the South Pole, which effectively detects neutrinos. Detection concretely means interaction with the detector.
In the case of extremely dense matter, such as during the supernova explosion, the interaction with neutrons is significant. That the mass of a neutrino is zero is also incorrect, there are several indications that the mass is extremely small, but not zero.

With the formula E = mc² you have to be careful, especially with regard to the meaning of m.
If the rest mass m_O is completely converted into energy, you get an amount of energy:

E = m_O c²

However, the total energy is also partly determined by the kinetic energy, due to any movement. Then you get:

E = root (m_O² c⁴ + p²c² )

or

E² = m_O² c⁴ + p²c²

If you then write:

E = mc² then m stands for the relatviistic mass, and we have:

E = mc² = m_O c² + p²c²

for a photon is m_O equals zero, but the momentum p is not, so the energy of a photon is

E = pc

and this is also equal to E = h.ν with ν the frequency and h the Plankck constant, and hence E = hc / λ with λ the wavelength.