You are correct that for a massive spinor, helicity is not Lorentz invariant. For a massless spinor, helicity is Lorentz invariant and coincides with chirality. Chirality is always Lorentz invariant.
Helicity defined
$$
\hat h = \vec\Sigma \cdot \hat p,
$$
commutes with the Hamiltonian,
$$
[\hat h, H] = 0,
$$
but is clearly not Lorentz invariant, because it contains a dot product of a three-momentum.
Chirality defined
$$
\gamma_5 = i\gamma_0 \ldots \gamma_3,
$$
is Lorentz invariant, but does not commute with the Hamiltonian,
$$
[\gamma_5, H] \propto m
$$
because a mass term mixes chirality, $m\bar\psi_L\psi_R$. If $m=0$, you can show from the massless Dirac equation that $\gamma_5 = \hat h$ when acting on a spinor.
Your second answer is closest to the truth:
The weak interaction couples only with left chiral spinors and is not frame/observer dependent.
A left chiral spinor can be written
$$
\psi_L = \frac12 (1+\gamma_5) \psi.
$$
If $m=0$, the left and right chiral parts of a spinor are independent. They obey separate Dirac equations.
If $m\neq0$, the mass states $\psi$,
$$
m(\bar\psi_R \psi_L + \bar\psi_L \psi_R) = m\bar\psi\psi\\
\psi = \psi_L + \psi_R
$$
are not equal to the interaction states, $\psi_L$ and $\psi_R$. There is a single Dirac equation for $\psi$ that is not separable into two equations of motion (one for $\psi_R$ and one for $\psi_L$).
If an electron, say, is propagating freely, it is a mass eigenstate, with both left and right chiral parts propagating.
You are looking for a unitary representation of parity on spinors. That it should be unitary can be seen from the fact, that parity commutes with the Hamiltonian. Compare this to time-reversal and charge conjugation, which anticommute with $P^0$ and hence need be antiunitary and antilinear. They involve complex conjugation.
As demonstrated parity transforms a $(\frac{1}{2},0)$ into a $(0,\frac{1}{2})$ representation. Hence it cannot act on any such representation alone in a meaningful way. The Dirac-spinors in the Weyl-basis on the other hand contain a left- and right-handed component
$$ \Psi = \begin{pmatrix}
\chi_L \\ \xi_R
\end{pmatrix} $$
As a linear operator on those spinors - a matrix in a chosen basis - it mixes up the spinor components. After what has been said before, left- and right-handed components should transform into each other. The only matrix one can write down that does this is $\gamma^0$. There could in principle be a phase factor. In a theory with global $U(1)$-symmetry this may be set to one however.
Edit:
Statements like $\chi_L \rightarrow P\chi_L=\chi_R $ for a Weyl-Spinor $\chi_L$ are not sensible. The Weyl-spinors are reps. of $\mathrm{Spin(1,3)}$, whereas $P\in \mathrm{Pin(1,3)}$. One cannot expect that some representation is also a representation of a larger group. Dirac-spinors on the other hand are precisely irreps. of $\mathrm{Spin(1,3)}$ including parity, which cannot act in any other sensible way than by exchanging the chiral components.
Think of what representation means. It's a homomorphism from a group to the invertible linear maps on a vector space. $$ \rho: G \rightarrow GL(V)$$
Particularly, for any $g\in G$ and $v\in V$, $\rho(g)v\in V$. Now set $V$ to be the space of say left-handed Weyl-spinors and $g=P\in\mathrm{Pin(1,3)}$ the parity operation. As you have shown above, the image of a potential $\rho(P)$ is not a left-handed Weyl-spinor, hence is not represented.
Best Answer
Everything depends on how your fields (vectors and spinors are fields in the classical theory, and when you quantize in QFT, they become operator-valued fields) transform when you make a Lorentz transform:
An scalar is a field that doesn't change at all: $\phi'(x') = \phi(x)$. Examples are the Higgs and pions.
A vector field is a field that transform like a relativistic four-vector $A'_\mu (x') = {\Lambda^\nu}_\mu A_\nu(x)$, where $\Lambda$ is a Lorentz transformation. Examples are the electromagnetic field (photons) and gluons.
An spinor field transform using a different set of matrices $$\psi'(x') = \exp\left[(-i \vec{\theta} \pm\vec{\eta}) \cdot \frac{\vec{\sigma}}{2}\right] \psi(x)$$ where $\vec{\theta}$ are the angles of rotation along the axes, $\vec{\eta}$ the rapidity and $\vec{\sigma}$ the Pauli matrices. As you can see, Pauli matrices are 2x2 matrices, so this transforms acts on objects with two components, the Weyl spinors. I have written two signs, $\pm$, because there are two types of transformations that act on two types of spinors: $-$ for left-handed spinors $\psi_L$ and $+$ for right-handed spinors $\psi_R$. But Weyl spinors have two problems: when you make a parity transformation ($\vec{r} \to -\vec{r}$), spinors change their handeness, and we know that QED and QCD are invariant under parity. And the other, as you say, is that Weyl fields must be massless.
The Dirac spinor solves both problems. It is just (in the chiral representation) a left-handed and a right-handed Weyl spinors side-by-side $$\psi = \begin{pmatrix} \psi_L\\ \psi_R \end{pmatrix}$$ The Dirac spinor can have mass (although massless Dirac spinors are fine). Electrons, muons, taus, neutrinos and quarks are described Dirac fields.
The Majorana spinor is a special Dirac spinor, where the left-handed and the right-handed parts are not independent. This relationship means that a Majorana particle is equal to its antiparticle. Therefore, the Majorana field has no electric charge. Although you only need one Weyl spinor to determine a Majorana spinor, Majorana fields still can have mass. It is conjectured that neutrinos might be Majorana particles (there are several experiments researching this).
So, where is spin? Angular momentum is a conserved quantity related to rotations. Whe you apply Nöther's theorem to a field, you get two terms: one depends on the movement of the particles (the orbital angular moment) and the other not (the spin). The spin part is related to the type of Lorentz transform that the field uses: in scalar fields there is no spin term (they have spin 0), in spinor fields it is a representation of rotations of dimension 2 (spin 1/2), and in vector fields a representation of rotations of dimension 3 (spin 1).