Whereas there has been extensive theoretical and experimental investigation of the properties of linear polymer chains in solution, there has been far less work on sheet-like polymers having 2D connectivity and 3D crumpled or collapsed shapes caused by thermal fluctuations, attractive self-interactions, or both. Sheet-like polymers arise in a variety of contexts ranging from self-assembled biological membranes (e.g., the spectrin network of red blood cells, microtubules, etc.) to nanocomposite additives to polymers (carbon nanotubes, graphene, and clay sheets) and polymerized monolayers. We investigate the equilibrium properties of this broad class of polymers using a simple model of a sheet polymer with a locally square symmetry of the connecting beads. We quantify the sheet morphology and the dilute-limit hydrodynamic solution properties as a function of molecular mass and sheet stiffness. First, we reproduce the qualitative findings of previous work indicating that variable sheet stiffness results in a wide variety of morphologies, including flat, crumpled or collapsed spherical, cylindrical or tubular, and folded sheets that serve to characterize our particular 2D polymer model. Transport properties are of significant interest in characterizing polymeric materials, and we provide the first numerical computations of these properties for sheet polymers. Specifically, we calculate the intrinsic viscosity and hydrodynamic radius of these sheet morphologies using a novel path-integration technique and find good agreement of our numerical results with previous theoretical scaling predictions.