Potentially. The obvious answer is careful and redpeated handwashng with one of the many commercikally available skin disiinfectants.
Not particularly. You state the scene as if this is in the past, so I assume it was treated. There is no reason to treat someone with this history as if they have EBOLA and will taint their environment. Almost any surface you touch in any environment, including stores, schools, etc will have some staph germs, some of which are aggressive nasty strains. Normal handwashing and monitoring is all that is needed.
Higher than usual. Premature infants are at higher risk for infection in general because their immune systems are weaker and less developed. In the nicu, strict vigilant hand washing is a must and is taken very serious to reduce this risk. At home with a premie, you should do the same. If the baby is breast feeding, there is added protection for the baby. Wash you hands regularly.
Antibiotics. Treatment varies depending on the age of your infant, and the location and extent of the infection. In an older infant, if an abscess is present, it most likely will require incision and drainage followed by oral antibiotics and proper wound care. Mrsa infections would require certain antibiotics for cure. In young infants, IV antibiotics may required. See your dr!