The spectral energy distributions and infrared (IR) spectra of a sample of obscured AGNs selected in the mid-IR are modeled with recent clumpy torus models to investigate the nature of the sources, the properties of the obscuring matter, and dependencies on luminosity. The sample contains 21 obscured AGNs at z=1.3-3 discovered in the largest Spitzer surveys (SWIRE, NDWFS, & FLS) by means of their extremely red IR to optical colors. All sources show the 9.7micron silicate feature in absorption and have extreme mid-IR luminosities (L(6micron)~10^46 erg/s). The IR SEDs and spectra of 12 sources are well reproduced with a simple torus model, while the remaining 9 sources require foreground extinction from a cold dust component to reproduce both the depth of the silicate feature and the near-IR emission from hot dust. The best-fit torus models show a broad range of inclinations, with no preference for the edge-on torus expected in obscured AGNs. Based on the unobscured QSO mid-IR luminosity function, and on a color-selected sample of obscured and unobscured IR sources, we estimate the surface densities of obscured and unobscured QSOs at L(6micron)>10^12 Lsun, and z=1.3-3.0 to be about 17-22 deg^-2, and 11.7 deg^-2, respectively. Overall we find that ~35-41% of luminous QSOs are unobscured, 37-40% are obscured by the torus, and 23-25% are obscured by a cold absorber detached from the torus. These fractions constrain the torus half opening angle to be ~67 deg. This value is significantly larger than found for FIR selected samples of AGN at lower luminosity (~46 deg), supporting the receding torus scenario. A far-IR component is observed in 8 objects. The estimated far-IR luminosities associated with this component all exceed 3.3x10^12 Lsun, implying SFRs of 600-3000 Msun/yr. (Abridged)
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