This work is devoted to studying a family of Middle Proximal Alternating Direction Method of Multipliers (MP-ADM) for solving multi-block constrained separable convex optimization. Such one-parameter family of MP-ADM combines both Jacobian and Gauss-Seidel types of alternating direction method, and proximal point techniques are only applied to the middle subproblems to promote the convergence. We analyze the global convergence as well as the sublinear convergence rate of the proposed algorithm in the ergodic sense. Moreover, its linear convergence is also established under the assumptions that the sub-differential of each component function in the objective function is piecewise linear and all the constraint sets are polyhedra. Our MP-ADM is later extended to a general version with positive-indefinite proximal terms. Compared with several state-of-the-art algorithms in the literature, the new MP-ADM is validated to be numerically efficient for solving a latent variable Gaussian graphical model selection problem arising from statistical learning.