Chirality of quantum spin Is quantum spin chiral?
If the answer is yes, then we can have four possibilities of quantum state in one direction of measure: Leftie pointing Up, leftie pointing Down, righty pointing Up, righty pointing Down.
About which two states says Pauli's exclusion principle? About opposite chirality or about opposite direction, or some combination of both?
When two particles are entangled? When they have the same chirality, direcrion, or some combination of chirality and direction?
 A: A spin is not chiral, but a particle with spin and velocity is.
Positive spin + Positive velocity -> Positive Helicity
Positive spin + Negative velocity -> Negative Helicity
Negative spin + Positive velocity -> Positive Helicity
Negative spin + Negative velocity -> Negative Helicity
So, in some sense, yes. You can have either helicity with either sign of spin depending on the sign of the velocity.
Pauli exclusion concerns the exchange of fermions in general, which may have spin, angular momentum, total angular momentum, position ... whatever ... you have a full set of quantum numbers, given by the symmetries of the system, that identify your fermion. Two fermions simply may not have the same full set. Context is necessary. Are they bound by a potential? Are they free?
In the condensed matter context, I think, you may take the helicity as a quantum number. In inversion symmetry broken materials, Dirac cones are forbidden, and Weyl cones become allowed. You may very well see that as a manifestation of "fermions may not have the same chirality at the same wave vector".
A: Chirality is the property that something does not look the same as its mirror image. In physics, the process of producing a mirror image is called parity.  When we apply a parity operation to a particle with a given spin, we get the same particle with the opposite spin. Therefore, spin has chirality. But that does not mean that spin and chirality are two distinct properties.
