Is there a name for the knob that's left at the point the fruit grows from? Also, what is best place the cut the stem when removing fruit, so new growth is created? I have a lot of the knobs left on my trees from fruit that dropped and I wonder if I should cut these knobs off?

Mark when grafting I often use the stems as scion wood and have found they have more buds than normal scion's from the limb. In my opinion these stems don't need to be removed.