In this paper, we study the relation between the finiteness properties of subgroups of direct products of 2-dimensional coherent right-angled Artin groups with their structure and the decidability of algorithmic problems. More precisely, we show that a finitely presented subgroup S of the direct product of 2-dimensional coherent RAAGs is virtually a nilpotent extension of a direct product. Moreover, if S is of type FP, then S is commensurable to a kernel of a character. We use these results to show that the multiple conjugacy problem and the membership problem are decidable for finitely presented subgroups of direct products of 2-dimensional coherent RAAGs. This work generalizes the results of Bridson, Howie, Miller and Short for free groups.