How much does a cut have to bleed, or how deep does it have to be, before stitches are necessary?

It depends. It varies based on where the cut is on the body, and if anything vital underneath the cut is also injured. Small cuts over areas that move (joints) or tight skin may be more important to stitch than a larger cut on the abdomen or between the joints on the arms and legs. Also, any cut that is difficult to stop bleeding needs to be seen. If the skin doesn't stay together as you move, get seen.