This work investigates the properties of stochastic quasi-Fejér monotone sequences in Hilbert spaces and emphasizes their pertinence in the study of the convergence of block-coordinate fixed point methods. The iterative methods under investigation feature random sweeping rules to select the blocks of variables that are activated over the course of the iterations and allow for stochastic errors in the evaluation of the operators. Algorithms using quasinonexpansive operators or compositions of nonexpansive averaged operators are constructed. The results are shown to yield novel block-coordinate operator splitting methods for solving structured monotone inclusion and convex minimization problems. In particular, the proposed framework leads to the design of random block-coordinate versions of the Douglas-Rachford and forward backward algorithms and some of their variants.