Is the number of nucleoli in a nucleus fixed?
I'll directly go to your fourth question, as all of them can be answered with a yes. This rather old mouse paper already got to the point. Sadly, only the abstract is freely available. Briefly, the number can be variable in the smallest unit you are interested in, the cell type. From the abstract:

The distribution of the number of nucleoli in many diploid cells
exhibited a mode of two or three nucleoli per nucleus, and a range
from 1 to 6 nucleoli.

In the paper, there are histogram of different cell types of mouse and all of them seem to be variable in their number of nucleoli.