It would be useful, if you'd point out the precise "open questions" regarding these evaluations that are of interest to you.
But since you mentioned Plamen Koev's work, I am sure you have tried out his matlab code for computing Hypergeometric functions of matrix argument. For the particular case of the normalization constant of the Bingham distribution, I am sure methods for approximating high-dimensional integrals numerically will prove to be effective, because, although scary looking, the normalization constant still has a very nice form:
\begin{equation*} {}_1F_1(\frac12,\frac p2, A) := \int_{S^{p-1}} e^{x^TAx}dx, \end{equation*} where the integration is wrt to the uniform distribution on the unit hypersphere $S^{p-1}$.
(PS: This integral has previously been discussed on MO, e.g., here in this question of L. Nicolaescu)