One first important point is that what is called "mass"in the question is not the physical on-shell mass. The physical on-shell mass is defined as a pole in the two-point function of the field, this is a non-perturbative defintion which makes clear that the physical on-shell mass does not depend on an energy scale. As indicated in the question, the mass considered is the $\overline{MS}$ mass, computed in a $\overline{MS}$ renormalization scheme, which depends on an energy scale and whose physical interpretation is the difference between the bare mass at the given energy scale and the on-shell mass.

1. Indeed, the fermion of mass $M$ does not decouple from the $\overline{MS}$-mass of $\psi$ when $M$ goes to infinity. This means that the $\overline{MS}$-mass of $\psi$ is "UV-sensitive", it depends on the physics at very high energies.

2. I think the explanation is a variant of the chiral symmetry (it would be exactly the most standard chiral symmetry for $\psi' = \psi$).

If $M=0$, the Lagrangian has the following symmetry, variant of the chiral symmetry: $\psi_L \mapsto e^{i \alpha} \psi_L$, $\psi_R \mapsto e^{-i \alpha} \psi_R$, $\psi^{'}_L \mapsto e^{-i \alpha} \psi^{'}_L$, $\psi^{'}_R \mapsto e^{i \alpha} \psi^{'}_R$. This symmetry forbids a mass term for $\psi$ or $\psi'$. This means that at the perturbative level, there will be no $\overline{MS}$ mass correction.

Now if $M \neq 0$, there will be some $\overline{MS}$-mass correction for $\psi$ but this correction should vanish in the limit $M$ goes to zero. This explains proportionality to $M$ and the absence of other terms proportional for example to the boson mass.

3. It is not so strange because it is not the physical on-shell mass which varies in this way but the $\overline{MS}$-mass.

Remark: the story of the $\overline{MS}$-mass corrections for massless fermions is of great phenomenological importance. The UV-sensitiveness of the correction (see 1.) means that it is "unnatural" to have massless fermions in some theory whithout having some symmetry explaining it: as the $\overline{MS}$-mass is the difference between some bare mass defined at some UV cut-off and the on-shell mass, having a on-shell mass equal to zero means having the bare mass being fine-tuned to compensate the $\overline{MS}$-corrections. One symmetry making massless fermions "natural" is chirality (see 2.). In the Standard Model, there are chiral gauge couplings making massless fermions "natural". This "explains" why there exists massless fermions or rather fermions light with respect to some "natural" scales (GUT, Planck scales), with masses coming only from some spontaneous breaking symmetry phenomenon (Higgs mechanism).