We use the dual perspective for optimal stopping, introduced by Davis and Karatzas, to derive pathwise conditions for commonly sought properties of optimal stopping problems. In particular, we show that the continuity of the value function, the continuity of the optimal stopping time, the continuity of the spatial derivative of the value function, and the existence of equilibria in a large class of stopping games can be derived from the path properties of the reward process and its Snell envelope. We provide several illustrative examples and applications.