Yes, in a way. The nucleus contains most of the cell's genetic material and is responsible for running most of its functions. (Sorry, I just had to change the previous answer. Hope it helps =] )
The brain cells, after the age of development (Usually 20 years) are locked in G0 or G Zero stage.
The nucleus
Yes. Somatic cells = Nonreproductive Cells
That is the name of the cell. Neurons are also called nerve cells.