Thank you, CrossRoads!
So, in other words we do need the 220 Ohm resistors between the shift register and the NDP6020P-ND gate. We also need pullup resistors between the gate and source of the NDP6020-ND and their value should be 5 KOhm.
As for the LED current limiting resistor, I understand now which components drop voltage and need to be taken into account. I guess your guess of 0.5 is from the NDP6020-ND. I've read the datasheet, not just once, but I can't figure out the voltage drop based on the graphs. Here is the excat datasheet of the NDP6020-NDs that I've bought from Farnell: http://www.farnell.com/datasheets/59594.pdf
The 3.3 voltage drop for the LEDs is a correct assumption.
One more important question: CrossRoads, in your image the Arduino had some additional things (capacitors, resistors) connected to some pins on the top and left, so it would seem that not only the MOSI, SCK, SS and GND pins are connected. Is that needed, or is that just some leftover from another circuit? If needed, what values would those be and what is their role?
Thank you very very much or your answer!