Final answer:
To treat the rash presented by the 10-year-old patient, which is indicative of impetigo, the most appropriate antibiotic is dicloxacillin. This antibiotic is suitable for treating staphylococcal infections, which the Gram stain suggests to be the cause of the impetigo.
Step-by-step explanation:
The patient's presentation of a non-healing rash with a progression from small, intensely itchy and erythematous areas to fluid-filled blisters, and finally to lesions in various stages covered with a golden-yellow crust, alongside the Gram stain showing Gram-positive cocci in clumps, strongly suggests a diagnosis of impetigo. Given the bacterial findings from the gram stain, the most appropriate treatment would be an antibiotic effective against staphylococci. From the options provided, dicloxacillin is commonly used for treating impetigo caused by staphylococcal infections. Dicloxacillin is preferred over the use of vancomycin, which is generally reserved for methicillin-resistant Staphylococcus aureus (MRSA) infections or patients who cannot tolerate penicillin-type antibiotics. Trimethoprim-sulfamethoxazole is also used for MRSA infections. Acyclovir would be indicated for viral infections such as herpes or varicella-zoster, which do not align with the presented symptoms or Gram stain results. Varicella zoster immune globulin (VZIG) is not appropriate, as the child's presentation does not fit with a primary varicella infection, and his immunizations are up-to-date.