Global distributions of the dayside magnetosheath proton density and temperature are reconstructed, using a multi-year pool of satellite observations and empirical data-constrained models with a flexible architecture. The models are based on data of Themis, Cluster, and MMS satellites and are driven by the upstream solar wind density, speed, proton temperature, and interplanetary magnetic field (IMF). In the simplest case of zero IMF and untilted geodipole, the models reveal the expected peak of plasma density and temperature in the subsolar area, with gradual decrease away from the Sun-Earth line. In more complex situations with nonzero IMF, the magnetic field draping and associated electromagnetic tensions, combined with the reconnection-induced plasma energization, streaming, and pile-up effects at the magnetopause, result in dawn-dusk and north-south asymmetries. The north-south asymmetry is dramatically enhanced during periods with large geodipole tilt to the terminator plane. The asymmetry is especially pronounced around the sunward (local summer) polar cusp, more directly exposed to the solar wind. In this case, the cusp's poleward boundary forms a local magnetic wall that brakes the particle flow and, for parallel orientation between the magnetosheath and magnetosphere tail lobe field, forms a pile-up area with enhanced density. In the opposite case of antiparallel alignment between the magnetosheath and lobe fields, the inflowing plasma is effectively drained away by reconnection. The obtained results are compared with those of gasdynamic theory and MHD simulations. In particular, a remarkable agreement is found with theoretically predicted density and temperature jumps across the subsolar bow shock.