We apply broken-symmetry density functional theory to determine isotropic exchange coupling constants and local zero-field splitting (ZFS) tensors for the tetragonal (Mn12BuAc)-Bu-t single-molecule magnet. The obtained parametrization of the many-spin Hamiltonian (MSH), taking into account all 12 spin centers, is assessed by comparing theoretical predictions for thermodynamic and spectroscopic properties with available experimental data. The magnetic susceptibility (calculated by the finite-temperature Lanczos method) is well approximated, and the intermultiplet excitation spectrum from inelastic neutron scattering (INS) experiments is correctly reproduced. In these respects, the present parametrization of the 12-spin model represents a significant improvement over previous theoretical estimates of exchange-coupling constants in Mn-12, and additionally offers a refined interpretation of INS spectra. Treating anisotropic interactions at the third order of perturbation theory, the MSH is mapped onto the giant-spin Hamiltonian describing the S = 10 ground multiplet. Although the agreement with high-field EPR experiments is not perfect, the results clearly point in the right direction and for the first time rationalize the angular dependence of the transverse-field spectra from a fully microscopic viewpoint. Importantly, transverse anisotropy of the effective S = 10 manifold is explicitly shown to arise largely from the ZFS-induced mixing of exchange multiplets. This effect is given a thorough analysis in the approximate D-2d spin-permutational symmetry group of the exchange Hamiltonian.