It is not only children, but adults, too. The root cause of a phobia is the fear, the definition of phobia, but it is because children do not know enough about the world, so instead of embracing and learning about it, they fear it, causing children to have phobias. It also may be due to past experiences that children have phobias, and those past experiences are fear-related.

The phobia of phobias is Phobophobia.

Phobias are an aspect of psychology, which is considered a science. So yes, phobias are a science of sorts

Freud said phobias were repressed conflicts, were as Watson said that phobias were learned

Honestly, you can have a phobia of anything! There is even a fear of phobias!