Atopic Dermatitis (Eczema): This condition is characterized by intense itching, small bumps, and swollen skin, which matches the symptoms described. The absence of fever and the distribution of the rash on the chest, back, arms, legs, and ears are also consistent with atopic dermatitis.

Allergic Contact Dermatitis: This could be due to an allergic reaction to something that came into contact with the skin, such as a plant, metal, or chemical. The symptoms of itching, small bumps, and swelling are consistent with this diagnosis.
Insect Bites: Multiple insect bites could cause small bumps and intense itching all over the body. The lack of fever makes this a plausible option.
Scabies: Although scabies typically cause more severe itching at night and burrows in the skin, it's possible that an early case or a case in a young child might present differently.

Bullous Pemphigoid: A rare autoimmune disease that can cause blistering of the skin and intense itching. Although less common, missing this diagnosis could lead to significant morbidity.
Stevens-Johnson Syndrome: A severe skin and mucous membrane disorder usually caused by an adverse reaction to medication or infection. The presence of swollen eyes and skin lesions could be an early sign, and missing this diagnosis could be fatal.

Mastocytosis: A condition where mast cells accumulate in the skin, causing itching, hives, and sometimes small bumps. It's less common but could explain the widespread nature of the symptoms.
Urticaria Pigmentosa: A type of mastocytosis that presents with brown patches on the skin that itch when stroked. It's rare and might not fully explain all symptoms but is worth considering in the differential diagnosis.