I assume you are not asking for a tutorial of the Dirac & Majorana equations or C, amply covered in standard texts like Li & Cheng, Schwartz, or Ramond’s “Journeys Beyond the Standard Model” —the horse’s mouth! You might prefer Klauber. I’ll just answer your narrower questions with the standard mnemonic.
First, go back to pre superKamiokande days, so, then without a weak singlet $\nu_R$ sterile neutrino, so neutrino Dirac masses —or a seesaw speculated upon in 1979.
So all neutrinos and antineutrinos have weak charge (doublet indices partnered with those of the charged leptons), conjugate to each other.
$\nu_L$ destroys a LH neutrino and creates a RH antineutrino.
Its hermitian conjugate, $\overline{\nu_L}$, creates a LH neutrino and destroys a RH antineutrino.
The SU(2)-symmetric charge current vertex (+h.c.) is then proportional to
$$
W^-_\mu \overline{e_L} \gamma^\mu \nu_L + W^+_\mu \overline{\nu_L } \gamma^\mu e_L ,
$$
where $e_L$ destroys a LH electron, and creates a RH positron, as before, and $\overline{e_L}$ creates a LH electron and destroys a RH positron.
Since $W_\mu^-$ destroys a $W^-$ and creates a $W^+$, the first term of this charge-conserving (and weak isospin conserving) interaction destroys a LH neutrino and creates a LH electron and a $W^+$, etc. The second term, likewise, destroys a RH antineutrino and creates a RH positron and a $W^-$, etc, for all line reversals of your choice.
So, your sweeping judgment that the weak vertices “only couple to LH fields” is unwarranted when the chirality-reversing bars sheltering antiparticles are considered. Now, since $\nu_L^c=C \overline{\nu_L}$ does what $\overline{\nu_L} $ does ; and $ \overline{\nu_L} ^c= -\nu_L C^{-1}$ does what $\nu_L $ does; and, mutatis mutandis for the electrons, the above weak vertex presents, perversely, as
$$
W^-_\mu e_L ^c\gamma^\mu \nu_L + W^+_\mu \nu_L ^c\gamma^\mu e_L ,
$$
where the evident transposes are understood, so that a c superscript does what an overbar does. Again, the first term destroys a LH neutrino and creates a LH electron and a $W^+$, or else creates a RH antineutrino and destroys a RH positron and a $W^-$; etc for all line reversals of your choice.
Because $\overline{\nu_L} (\nu_L)^c$ is simply the definition of a Majorana mass, not the Dirac mass term; it preserves Lorentz invariance, but fails to obey the Dirac equation, as all QFT texts detail. It violates lepton number, as it destroys a LH neutrino and a RH antineutrino, or else creates a RH neutrino and a RH antineutrino. It is a source or sink term for lepton umber 2. (A corresponding term for the electron would also violate lepton number by 2, but, crucially electric charge by 2, as well, and of course would fail to be an isosinglet.) Therefore, if you looked at the weak indices, suppressed, you’d see that this term cannot be a weak isosinglet, so it is forbidden. (Except in the unrenormalizable, dimension 5, “Weinberg term”, where each neutrino’s index is saturated to a singlet by the neutral component of a Higgs each, whose v.e.v. sinks into the vacuum taking away one total unit of weak isospin. I gather this is what you have in mind for your next question.)
Enter sterile EW singlets (indexless) $\nu_R$ and the seesaw. Now the superheavy N will be mostly $\nu_R$ with a tiny contamination $O(m_D/M)$ component of (indexed!) $\nu_L^{~c}$, which, ipso facto, as you are surmising, does couple to the Ws. Nevertheless, this is a minuscule component of order 100GeV/$10^{15}$GeV ~ $10^{-13}$, so, negligible. The major handle on this mixing, instead is through lepton number violation involving the other, the light eigenvector of the seesaw mass matrix, not N; it will enter into the neutrinoless double β decay.
