In this paper, we consider a three node, two-way relay system with digital network coding over static channels where all link gains are assumed to be constant during transmission. The aim is to minimize total energy consumption while ensuring queue stability at all nodes, for a given pair of random packet arrival rates. Specifically, we allow for a set of transmission modes and solve for the optimal fraction of resources allocated to each mode, including multiaccess uplink transmission mode and network coding broadcasting mode. In addition, for the downlink, we find the condition to determine whether superposition coding with excess data over the better link and network coded data for both users is energy efficient and the corresponding optimization is formulated and solved. To tackle the queue evolution in this network, we present a detailed analysis of the queues at each node using a random scheduling method that closely approximates the theoretical design, through a two-dimensional Markov chain model.