This paper studies distributed quantized weight-balancing and average consensus over fixed digraphs. A digraph with non-negative weights associated to its edges is weight-balanced if, for each node, the sum of the weights of its outgoing edges is equal to that of its incoming edges. We propose and analyze the first distributed algorithm that solves the weight-balancing problem using only quantized (one-bit) information among nodes and simplex communications (compliant to the directed nature of the graph edges). Asymptotic convergence of the scheme is proved and a convergence rate analysis is provided. Building on this result, a novel distributed algorithm is proposed that solves the average consensus problem over digraphs, using, at each iteration, only two-bit simplex communications between adjacent nodes - one bit for the weight-balancing problem, the other for the average consensus. Convergence to the average of the real (i.e., unquantized) node's initial values is proved, both almost surely and in mean square sense. Finally, numerical results validate our theoretical findings.