1)you have seen the result on CRO.
Full wave rectifer gives continuos pulses while in half wave rectifier there is a constant break(0V) two consecutive pulses.
It depends on the requirement which type of pulse train you want.
3) In a wave, every point vibrates about its mean postion.Phase is used to denote the relative displacement from a postion taken as standard(usually it is mean position). If two points always have the same displacement, they are in the same phase.If at some instant, two points have the same displacement, they are not necessarily in same phase because they may be moving in opposite directions, hence will not be in the same phase at the next instant.