Multiple LED PWM - LED on when it shouldn't be (PWM at 0)

Well those FETs are wrong for a start. They need 10V on the gate to fully turn them on and you only get 5V from the arduino.

However your description would suggest there is something else wrong as well. Perhaps it is not wired the same as the schematic. If it were the problem you should get is that the LEDs are not coming on fully.