To better understand the self-assembly of small molecules and nanoparticles adsorbed at interfaces, we have performed extensive Monte Carlo simulations of a simple lattice model based on the seven hard "tetrominoes", connected shapes that occupy four lattice sites. The equations of state of the pure fluids and all of the binary mixtures are determined over a wide range of density, and a large selection of multicomponent mixtures are also studied at selected conditions. Calculations are performed in the grand canonical ensemble and are analogous to real systems in which molecules or nanoparticles reversibly adsorb to a surface or interface from a bulk reservoir. The model studied is athermal; objects in these simulations avoid overlap but otherwise do not interact. As a result, all of the behavior observed is entropically driven. The one-component fluids all exhibit marked self-ordering tendencies at higher densities, with quite complex structures formed in some cases. Significant clustering of objects with the same rotational state (orientation) is also observed in some of the pure fluids. In all of the binary mixtures, the two species are fully miscible at large scales, but exhibit strong species-specific clustering (segregation) at small scales. This behavior persists in multicomponent mixtures; even in seven-component mixtures of all the shapes there is significant association between objects of the same shape. To better understand these phenomena, we calculate the second virial coefficients of the tetrominoes and related quantities, extract thermodynamic volume of mixing data from the simulations of binary mixtures, and determine Henry's law solubilities for each shape in a variety of solvents. The overall picture obtained is one in which complementarity of both the shapes of individual objects and the characteristic structures of different fluids are important in determining the overall behavior of a fluid of a given composition, with sometimes counterintuitive results. Finally, we note that no sharp phase transitions are observed but that this appears to be due to the small size of the objects considered. It is likely that complex phase behavior may be found in systems of larger polyominoes.