Various high-order Hermite-Gaussian (HG) modes with high repetition rates and high peak powers are systematically generated by designing the cavity configuration to satisfy the criterion of the passive Q-switching. For the HGm,0 modes with the order m  =  1-9, the pulse repetition rate can exceed 100 kHz with peak power higher than 0.3 kW. For the HGm,m modes with the order m  =  1-10, the pulse repetition rate can be up to 37 kHz with peak power higher than 0.35 kW. Furthermore, the high-order HG beams is transformed by using an astigmatic mode converter to generate various structured lights with optical vortices. Experimental patterns of the transformed high-order HG beams in the propagation are theoretically analyzed and the phase structures are numerically manifested.