The weight of the axe head does play a factor. However a heaver head is harder to swing, takes longer to accelerate, will have a lower top speed, and caused the robot to jump more when compared to a lighter head. Its a trade off between mass and speed and it will depend on what kind of axebot you want.

A 3-Phase driver is an ESC, and pretty much all hobby brushless motors are three phase. Rory is more or less right about the issues with brushless and how they technically have no start up torque at all. This can be partly solved with sensors and smarter Firmware on the ESC's; you may have heard of SimonK, BL heli and VESC's, but as you will have seen in Robot Wars, reliable and affordable brushless controllers that can match the very mature brushed setups people use are not quite ready. This is less of an issue in FW's and lighter classes but the fundamental issues with brushless motors are still present.