Does the concept of both helicity and chirality make sense for a massive Dirac spinor? Does the concept of both helicity and chirality make sense for a massive Dirac spinor?
A massive electron in the chiral basis is written as a column made up of $\psi_L$ and $\psi_R$. What is the significance of this?
These cannot be decoupled from each other due to the presence of the mass term in the Dirac equation,
and we know electrons will look either left handed or right handed depending upon from which frame we are looking at them.
But what is then the meaning of $\psi_L$ and $\psi_R$ separately?
 A: It is important to differentiate between helicity and chirality. Helicity is the spin angular momentum of a particle projected onto its direction of motion. For a massive particle this quantity is frame dependent. Furthermore, since angular momentum is conserved, as a particle propagates helicity is conserved.
On the other hand, chirality is an innate property of a particle and doesn't change with frame. However, the mass term for a Dirac particle is,
\begin{equation}
-m(\psi_L^\dagger \psi_R + \psi_R^\dagger\psi_L)
\end{equation}
(in this notation the Dirac spinor is $\Psi= (\psi_L , \psi_R) ^T $). This term can be thought of as an interaction term in the Lagrangian which switches the chirality of a particle (e.g. a left chiral particle can spontaneously turn into a right chiral particle)
For a massless particle, chirality is equal to helicity.

With that background we can finally address your questions.

*

*Both helicity and chirality definitely make sense for a massive
Dirac spinor. However, that doesn't mean that a Dirac spinor is a
helicity and chirality eigenstate. In the same sense that energy
makes sense for a particle, but it may not be an energy eigenstate.

*As you mention the left chiral and right chiral fields can't be
decoupled from each other due to the mass term. The mass term can
always switch a right handed field to a left handed field and vice
versa.

*As I said above, the helicity of an electron is indeed frame
dependent. So it may look like a left or right helicity electron
depending on the frame, however its chirality is not frame
dependent.

*If we write the Dirac Lagrangian in terms of chirality eignestates
then we have, \begin{equation} {\cal L} _D = i \psi _L ^\dagger 
\sigma ^\mu \partial _\mu \psi _L + i \psi _R  ^\dagger \bar{\sigma}
^\mu \partial _\mu \psi _R - m \psi _L ^\dagger \psi _R - m \psi _R
^\dagger \psi _L  \end{equation} Then we can think of $\psi_L$ (left
chiral particle) and $\psi_R$ (right chiral particle) as two
different particles that can turn into each other spontaneously
through a mass term. Putting them together, into a Dirac spinor
masks this property. However, they are still well defined
separately.

