@Willie the typo is corrected. Thanks

I've just corrected the wording.

Maybe a better question could be if there is a way to deform minimal surfaces (mean curvature equals zero) to obtain at the limit that list of minimal cones. Maybe by only looking at the Weierstrass-Enneper representation it shouldn't be so difficult to produce examples of minimal surfaces that deforms into those cones, I wonder if someone has already done that or if this is totally wrong.

On the other hand, is there a link between $c$-cyclically monotone sets (from optimal transport theory) and rectifiability?

I have not yet read the papers you mentioned above but I'm studying a course of Nicola Gigli on the subject, I will read those papers very soon. Do you have the reference of Bonnet-Myers in $CD(K,N)$?

For example, can we obtain Bonnet-Myers theorem just using the convexity of the entropy on $\mathcal{P}^2(M)$?. By analogous I mean similar.