I think that the referenced paper has all you might want (it is obviously related to the Derevnyn-Mednykh paper, but is considerably more extensive). They do not specifically talk about orthoschemes, but their results specialize nicely.
A. D. Mednykh and M. G. Pashkevich, "Elementary formulas for a hyperbolic tetrahedron", Sib. Math. J. 47 (2006), 687-695.
