This paper is a companion to our recently published semiclassical formalism for treating time-dependent Hamiltonians [J. Chem. Phys, 105, 4094 (1996)], which was applied to study the dissociation of diatomic ions in intense laser fields. Here two fundamental issues concerning this formalism are discussed in depth : conservation principles and coherence. For time-dependent Hamiltonians, the conservation principle to apply during a trajectory hop depends upon the physical origin of the electronic transition, with total energy conservation and nuclear momentum conservation representing the two limiting cases. It is shown that a]applying an inappropriate scheme leads to unphysical features in the kinetic energy of the dissociation products. A method is introduced that smoothly bridges the two limiting cases and applies the physically justified conservation scheme at all times. It is also shown that the semiclassical formalism can predict erroneous results if the electronic amplitudes for well-separated hops are added coherently. This is a fundamental problem with the formalism which leads to unphysical results if left unattended. Alternative schemes are introduced for dealing with this problem and their accuracies are assessed. Generalization of the well-known Landau-Zener formula to the time-dependent Hamiltonian case is derived, which allows one to significantly decrease the computational overhead involved with the numerical implementation of the semiclassical method. Finally, we show that in strong-field molecular dissociation a trajectory can ＇＇surf" a moving avoided crossing. In this case the hopping probability is a sensitive function of the interference between two closely spaced avoided crossing regions.