Is there any immunity after staphylococcal infection?

No. There is no ability for your body to form full protection from further staph infections in the future. In fact, your body is covered in staph on an ongoing basis every day (so is everyone else's) and staph is supposed to be there. Its just that sometimes, staph takes over and causes infections. You are still at risk of that happening again.
Yes. We do form antibodies but they are not very efficient since antibodies are better at viruses, cancer, and fungus. The white blood cells fight the bacterial infections but they do not retain any memory cells which is why re infection occurs.