We convexify the second-stage of two-stage stochastic programs (2SPs) with binary recourse using binary decision diagrams (BDDs) to make the problem amenable to Benders decomposition algorithm. More specifically, we first generalize an existing BDD-based approach to allow settings where logical expressions of the first-stage solutions enforce constraints in the second stage. We also propose a complementary problem where second-stage objective coefficients are impacted by logical expressions of the first-stage decisions, and develop a distinct BDD-based algorithm to solve this novel problem class. We extend this work by incorporating conditional value-at-risk and we propose, to our knowledge, the first decomposition method for 2SPs with binary recourse and a risk measure. We apply these methods to a novel stochastic dominating set problem and present numerical results to demonstrate the effectiveness of the proposed methods.