The pure translation (TR) imaginary-frequency (or unstable) instantaneous normal modes (INM), which we have proposed as representative of barrier crossing and diffusion, are obtained for seven densities and eight temperatures of supercooled and near-melting liquid CS2 via computer simulation. The self-diffusion constant D, with a range of over two decades, has been determined previously for these 56 states [Li and Keyes, J. Chem. Phys. 111, 328 (1999)], allowing a comprehensive test of the relation of INM to diffusion. INM theory is reviewed and extended. At each density Arrhenius T-dependence is found for the fraction f(u) of unstable modes, for the product (u)f(u) of the fraction times the averaged unstable frequency, and for D. The T-dependence of D is captured very accurately by f(u) at higher densities and by (u)f(u) at lower densities. Since the T-dependence of (u) is weak at high density, the formula D proportional to (u)f(u) provides a good representation at all densities; it is derived for the case of low-friction barrier crossing. Density-dependent activation energies determined by Arrhenius fits to (u)f(u) are in excellent agreement with those found from D. Thus, activation energies may be obtained with INM, requiring far less computational effort than an accurate simulation of D in supercooled liquids. Im-omega densities of states, , are fit to the function a(T)omega exp[-(a(2)(T)omega/root T)(a3(T))]. The strong T-dependence of D, absent in Lennard-Jones (LJ) liquids, arises from the multiplicative factor a(T); its activation energy is determined by the inflection-point energy on barriers to diffusion. Values of the exponent a(3)(T) somewhat greater than 2.0 suggest that liquid CS2 is nonfragile in the extended Angell-Kivelson scheme for the available states. A striking contrast is revealed between CS2 and LJ; a(3)--> 2 at low-T in CS2 and at high-T in LJ. The INM interpretation is that barrier height fluctuations in CS2 are negligible at low-T but grow with increasing T, while the opposite is true for LJ.