In a representation independent way, charge conjugation matrix can be defined as, C-1γμC=-γμT. If you plug in gamma matrices in Majorana representation (which is purely imaginary) in this basic definition you should get explicit form of C matrix. A charge conjugated spinor should reduce to ψc=-iγ2ψ*=ψ.

Here C is indeed an operator which acts on Dirac spinors (four component objects) and which can be represented by a 4x4 matrix. C does not commute with electric charge operator (or any other charge such as B, L,..). It is therefore not possible to have a simultaneous eigenstate of charge conjugation operator and electric charge operator. States without having any charges can be eigenstates of C, with eigenvalues plus or minus unity, known also as C parity.

Now left handed Majorana mass is ν̅LνLc and relabeling L→R yields right handed Majorana mass term. Let us denote these by mL and mR and the Dirac mass as mD, then, mν= mL + m2D / mR. In case you are studying a three generation case then replace mL, mR, mD by 3x3 complex matrices written in generation space and m2D by mD†mD.