Copulas are multi-dimensional probabilistic functions that play a crucial role in modeling complex dependence structures between random variables. The theory and applications of the three-dimensional case have attracted considerable interest in recent years. This article discusses recent developments in this specific topic and innovates in some aspects. More precisely, the first part proposed to fill certain theoretical gaps concerning a modern copula called the "product Ali-Mikhail-Haq mixed" copula and to complete its knowledge. Among these gaps, the possible values of the involved parameter that make it valid are revisited. The corresponding Spearman rho is also re-calculated. Mathematical proofs and graphic work are given. The second part presents a result on generalized convex mixed copulas and shows how the product Ali-Mikhail-Haq mixed copula can be considered as a specific example. The novelty of this result is that it is general in the dimensional sense and presents simple conditions to guarantee the validity of the resulting copula. Based on it, a new variant of the three-dimensional Ali-Mikhail-Haq copula is given. In summary, for the first time, some existing three-dimensional copulas are analyzed from the point of view of generalized convex mixture copulas in multiple dimensions, thus opening new horizons for dependence modeling.