We introduce surface fluctuating hydrodynamics approaches for investigating transport and fluid-structure interactions within curved fluid interfaces arising in soft materials. We focus particularly on lipid bilayer membranes and phenomena involving drift-diffusion dynamics of interacting proteins and microstructures. We show how a mesoscale stochastic description of the mechanics can be formulated (SPDEs) accounting for geometric contributions, hydrodynamic coupling, and thermal fluctuations. The underlying stochastic equations (SPDEs) pose challenges for use in practice, including, (i) a need for accurate and stable discretizations of geometric terms and differential operators on the surface, (ii) techniques for hydrodynamics handling surface incompressibility constraints, and (iii) stiffness from rapid time-scales introduced by the thermal fluctuations. We show how practical spectral methods and meshfree computational approaches can be developed for simulations over long spatial-temporal scales. We then present results of investigations of the role of geometry in hydrodynamic transport and the collective drift-diffusion dynamics of inclusions within curved membranes and other systems.