We have recently introduced a simple model which contains the basic features of phase separation in cross-linked polymer gels: a stretched elastic network of Lennard-Jones particles [Peleg, O.; et al. EPL 2007, 77, 58007]. While the original work clearly demonstrated the phenomenology of the model at a limited set of system parameters, this paper explores quantitative details and gives answers to many of the questions which remained open. We define and analyze an order parameter, explore the phase diagram, and identify the range of elastic constants in which microphase separation is observed and analyze the microphase separation patterns. We discuss the microscopic origin of the formation of linear rather than globular filaments as well as monitor and quantify the reorganization of the network, not only in equilibrium but also its dynamics upon applying a shear deformation. These simulations provide LIS with the elastic properties, most importantly, the shear modulus, whose magnitude is shown to correlate with the order parameter.