The washer isn't absolutely necessary, but I'd prefer it to be there. You could try to find a headset with a lower stack height, or even machining the locknut a bit to be able to install a washer. Some (me included) think it's better to file the tab from the washer, as it often messes up the steerer threads.
The steerer should turn smoothly from the start.
Cutting the steerer tube too short is a serious error for a professional mechanic.