Does extension of flaps 1, 2, 5 or 10 increase descent rates? Let's say you need greater descent rates than you can get with the full use of the speed brake in an airliner. Does it help to use the above flap positions in getting greater descent rates?
It depends what you intend to do with the speed. If all you are looking for is absolute descent rate:  a high-speed, spoilers out and high speed dive will get you the best rate.

What you are usually interested in a normal approach however is not descent rate (vertical speed) as much as descent angle. In this respect, all/any flaps will decrease the L/D ratio, which is going to increase you descent angle. Each stage of flaps will have increasingly higher descent angles, but mind you at a lower speed.