We theoretically investigate dynamics of dark pulse and Raman-Kerr microcombs generation influenced by higher-order effects, including high-order dispersion (HOD), stimulated Raman scattering (SRS) and self-steepening (SS) effects in silicon microresonators. These three effects cause the delay of dark pulse individually, or interact with each other to alter the drift velocity and direction of pulses. HOD effect can change pulse shift direction and even cause bifurcation. The temporal drift induced by SS or SRS effects could be balanced by the simultaneous third-order dispersion (TOD) engineering. In spectral domain, stable Raman-Kerr frequency comb will be generated due to the competition between strong SRS and Kerr effects. The Raman comb components are suppressed when HOD effect coexists, while SS effect has ignorable effect on the distribution of the Raman comb. Furthermore, the SS effect will increase the total energy of the spectrum by shifting the dispersive wave (DW) generation to the longer wavelength side. Our findings could deepen the understanding of intracavity nonlinear dynamics and provide theoretical guidance to precisely control the stabilization of dark pulse and the generation of broadband mid-infrared (MIR) microcomb.