The movement is a called a reflex, which is a sudden muscle movement in response to a stimuli. The reflex you are describing is a reflex known as a plantar reflex, which is present in infants and children but goes away before adulthood (ideally).
A scenario where a cell may be needed to perform a form of endocytosis is when transporting large molecules. A scenario where a cell may be needed to perform a form of exocytosis is when releasing the large molecule from the cell.
