The electron and neutrino spin Can you please clarify some basic notions about spin?
Can you please explain what is intended when they say that the spin of a neutrino is left-handed and equal to -1/2?

Does it mean its angular momentum is equal in direction and strength to the spin of the electron ( $h /4\pi$), even though its mass is about one million times smaller? If so, is there an explanation for that? Is there any experimental direct evidence of that magnitude?
Also, since the electron g-factor is 2, does that mean that its value is $h /2 \pi$ , equal to the orbital angular momentum in the hydrogen atom?
 A: The spin of a fundamental fermionic particle always has the absolute value $\frac{\hbar}{2}$. This does not relate to its orbital angular momentum, it's just what the spin of a fundamental fermion is.
It now turns out that, for massless particles, there is the notion of helicity (see also What is polarisation, spin, helicity, chirality and parity?), which is the projection (i.e. the relative direction) of spin and momentum.
In your picture, someone decided to orient the axis along which one component of the spin is measured parallel to the particles momentum, so that $+1/2$ spin corresponds to the spin being parallel to the linear momentum, and $-1/2$ to them being anti-parallel. In situations where "neutrinos are massless" is a good approximation, this picture is Lorentz-invariant and indeed a good description for the otherwise rather technical notion of chirality.
The electron cannot correspond to an anti-neutrino because the latter has zero electrical charge. Spin is only one of many quantum numbers.
A: 
My question is about the spin of the neutrino, and the evidence (or speculative arguing) for its magnitude which is not proportionate to its mass. 

You are dealing with "things" that are supposed to be point particles, so they should not, in classical terms, have a mass in the first place. The math takes over from common sense notions on this scale, and the spin of the neutrino has to be 1/2 to conserve quantum numbers. The mass is the link between the classical world, in which we expect it to be a certain value, and the quantum world, in which the particles are not real in a way we understand. 
I hopefully will not annoy you because you already know this. Because "spin" is an analogy, with absolutely nothing to do with ordinary soccer ball type spin, you could have asked the same question about the top quark, which has a huge mass but still has spin angular momentum of 1/2, because that is what we define it to have. 
So comparing the mass of one point object (which classically should not have a mass) with another point object with lower mass, and expecting a different angular momentum will not make sense in ordinary terms, but this is not the ordinary world you are asking about.
