How common is it for a staphylococcal infection to become septic?

Rare. But it can happen easily depending on the type of strep. Some are more virulent.
It depends. There are multiple kinds of staphylococcal bacteria strains. Whether a person develops sepsis is dependent on the type of staph strain, as well as the overall condition of the patient. For example, even a "weak" staph strain can cause sepsis in somebody whose immune system is weak. We all are exposed to staph virtually every day. It is impossible to quote you a percentage for your question.