When undergoing cytokinesis, animal cells and plant cells are pretty close to the same process. The only thing that is different is how the daughter cells were formed. In plant cells, the future cell wall, called the cell plate, divides the cytoplasm, which eventually forms the two new daughter cells. In animal cells, they don't have a cell wall, so they had to find a different way to do it. I am not 100% confident about this answer, but I am pretty sure it is right. A ring gets formed under the cell membrane of the cell, and creates a thing known as a furrow. The furrow keeps growing throughout the two cells, until it eventually gets super tight and pinches off the two cells.
So for the second question, genes and enzymes control the way that cells are produced and monitored. This goes well unless some kind of gene fails. It acts like a chain. So if a gene fails, then that fails the activation of the enzymes, and the failing of the enzymes leads to the uncontrolled population and growth of new cells. Without the enzymes to monitor what cells need to be created and when, cells will be created too rapidly. This rapid growth of cells lead to something called cancer, because of the uncontrolled cell division that was caused by the failing of enzymes.
That was a lot of information, so I hope it all makes sense! Ask questions in the comments if you don't completely understand something, and I will try my best to answer them. Hope you do well in biology! ;)