There is a general agreement that the conspicuous extranuclear X-ray, optical-line, and radio-contiuum emission of starbursts is associated with powerful galactic superwinds blowing from their centers. However, despite the significant advances in observational studies of superwinds, there is no consensus on the nature of the emitting material and even on the emission mechanisms themselves. This is to a great extent a consequence of a poor understanding of dynamical processes in the starburst superwind regions. To address this issue, we have conducted two-dimensional hydrodynamical simulations of galactic superwinds. While previous similar studies have used a single (disk) component to represent the ISM of the starburst galaxy, we analyze the interaction of the wind with a two-component disk-halo ambient interstellar medium and argue that this two-component representation is crucial for adequate modeling of starbursts. The emphasis of this study is on the geometry and structure of the wind region and the X-ray emission arising in the wind material and the shocked gas in the disk and the halo of the galaxy. The simulation results have shown that a clear-cut bipolar wind can easily develop under a range of very different conditions. On the other hand, a complex 'filamentary' structure associated with the entrained dense disk material is found to arise within the hot bubble blown out by the wind. The flow pattern within the bubble is dominated equally by the central biconic outflow and a system of whirling motions r elated to the origin and development of the 'filaments'. The filament parameters make them a good candidate for optical-emission-line filamentary gas observed in starburst halos. We find that the history of mass and energy deposition in the starburst region of the galaxy is crucial for wind dynamics. A 'mild' early wind, which arises as a result of the cumulative effect of stellar winds from massive stars, produces a bipolar vertical cavity in the disk and halo gas without strongly affecting the gaseous disk, thus creating conditions for virtually free vertical escape of the hot gas at the later, much more violent supernova-dominated phases of the starburst. We calculate the luminosity, mass, and effective temperature of the X-ray emitting gas in the 'soft' (0.1 to 0.7 keV, 0.7 to 2.2 keV, and 0.1 to 2.2 keV) and 'hard' (1.6 to 8.3 keV) energy bands and estimate the contribution of different gaseous components to the X-ray flux in these bands. Analysis of these parameters enables us to make conclusions regarding the nature of the X-ray-emitting material. We have inferred that the bulk of the soft thermal X-ray emission from starbursts arises in the wind-shocked material of the disk and halo gas rather than in the wind material itself. This enables us to predict that the integrated soft X-ray spectra of starbursts need not show an overabundance of heavy elements which are believed to be produced copiously in the centers of starbursts. Unlike soft X-ray emission, the hard component of thermal X-ray emission is found to originate in the wind material ejected from the starburst region. However, the derived ratio of hard-to-soft X-ray luminosities is too small compared to that observed in starbursts. We conclude therefore that the observed hard X-ray emission of starbursts is probably not associated with the thermal emission of hot wind or ambient shocked gas. Typical temperatures of the bulk of the soft X-ray-emitting material in our very different models have been found to agree well with the ones estimated on the basis of the ROSAT data for the soft component of X-ray emission of nearby starbursts. We predict that temperatures of the extranuclear soft X-ray-emitting gas in starburst galaxies with heavy element abundances near solar should be close to T(sub Xs = 2 to 5 x 10(exp 6)K.