Does not eating and exercise make your period start early?

No, if anything the opposite is true.

You only menstruate if you ovulate, and you only ovulate if your body is physically healthy enough to cope with pregnancy - if a woman was to exercise too much or starve herself then the body will try to prevent pregnancy by preventing or delaying ovulation, thus they would either miss a period or experience a late period.

Thus why girls who suffer from eating disorders like anorexia, or professional athletes, often loose their periods and can lose their ability to have children permanently.