We address two sets of fundamental problems in the context of equilibrium problems: (i) Of these, the first pertains towards the development of existence statements for stochastic variational inequality and complementarity problems. A naive application of traditional deterministic existence theory requires access to closed-form expressions of the expectation-valued map; instead, we develop a framework under which verifiable a.s. requirements on the map allow for making existence claims in a broad set of monotone and non-monotone regimes. Time permitting, we consider how such avenues can allow for building a sensitivity theory for stochastic variational inequality problems. (ii) Our second question considers the development of synchronous, asynchronous, and randomized inexact best-response schemes for stochastic Nash games where an inexact solution is computed by using a stochastic gradient method. Under a suitable spectral property on the proximal best-response map, we show that the sequence of iterates converges to the unique Nash equilibrium at a linear rate. In addition, the overall iteration complexity (in gradient steps) is derived and the impact of delay, asynchronicity, and randomization is quantified. We subsequently extend these avenues to address the distributed computation of Nash equilibria over graphs in stochastic regimes where similar rate statements can be derived under increasing rounds of communication and variance reduction.