Gutttate Psoriasis: This condition is often triggered by a streptococcal infection, but given the throat culture was positive for Staphylococcus, it's still a consideration due to the rash's description on palms, soles, and body, which is consistent with guttate psoriasis. However, the presence of Staphylococcus might lean more towards other diagnoses.

Impetigo: Given the recent throat culture positive for Staphylococcus and the vesicular rash, impetigo is a strong consideration. It's a highly contagious skin infection that can spread from the face to other parts of the body, including arms, palms, and soles.
Allergic Contact Dermatitis or Irritant Contact Dermatitis: Although less likely given the description, if the child has been exposed to new substances or has a history of allergies, this could be a consideration. The distribution on palms and soles could suggest an allergic or irritant reaction.
Pityriasis Rosea: This condition often starts with a herald patch followed by a widespread rash, but it doesn't typically involve the palms and soles as prominently as described. However, it's still a possibility, especially if the rash has evolved.

Staphylococcal Scalded Skin Syndrome (SSSS): Although the child is on cephalexin, which is effective against Staphylococcus, SSSS is a serious condition that can occur, especially in children, and is caused by staphylococcal toxins. It presents with widespread skin lesions and can be life-threatening if not promptly treated.
Toxic Epidermal Necrolysis (TEN) or Stevens-Johnson Syndrome (SJS): These are severe skin conditions usually triggered by medications. While cephalexin is less commonly associated with these conditions compared to other antibiotics, it's crucial not to miss them due to their high morbidity and mortality.

Kawasaki Disease: This condition can present with a rash, fever, and other systemic symptoms. Although it's rare and typically involves more than just a skin rash (including fever, conjunctivitis, and lymphadenopathy), it's a diagnosis that should be considered in the differential due to its potential severity and the need for prompt treatment.
Pemphigus Vulgaris: An autoimmune disease that can cause blistering of the skin and mucous membranes. It's rare in children and would be an unusual diagnosis, but it should be considered if other, more common conditions are ruled out.