Is it possible for a staph infection to become resistant to antibiotics?

Yes. Unfortunately almost all bacteria are able to develop resistance to an antibiotic. Particularly when used unnecessarily.Check with your doctor if this is a concern.
Yes . The organism staph aureus is notorious for morphing in to a resistant strain. Currently the two groups of staph that are of concern are h-mrsa and c-mrsa whose origins are hospital acquired and community acquired respectively. Besides the source of infection being different, they also have different antibiotic sensitivity, the hospital acquired type being the most resistant form.