We're spending hours with these questions, among others, so I should better ask whether someone knows the full answers.
Heterotic string theory apparently has the same divergent high-order structure of generic amplitudes that seem to behave as "(2h)!" for large genera "h"; the factor comes from the leading behavior of the volume of moduli spaces of genus "h" Riemann surfaces. In type II string vacua, this can be used to estimate the magnitude of the first non-perturbative corrections to the amplitudes; the result - comparable to the minimal, namely "1/g"th term in the expansion - is "exp(-C/g)" where "g" is the closed string coupling. These effects arise from D-instantons and D-branes in the loops, generally speaking.
In heterotic string theory you seem to get the same behavior but there are no D-branes whose tension or mass or action goes like "C/g". There are perhaps non-supersymmetric D2-branes or other D-branes in heterotic string theory. What is their lifetime? What is the minimum allowed lifetime for these objects that still allows one to approximate them by stable objects and derive that they also imply the "(2h)!" divergent structure of the amplitudes? Is it just about the comparison of the imaginary and real parts of the actions? It seems to be the case. When one tries to find the genus "h" for which the contribution is minimal, one may generically obtain a complex value of "h"; the "saddle" is then away from the real "h" axis. Is it indeed the case for some particular amplitudes?