If no fungus or bacteria infection, what may cause dandruf? can allergies cause dry skin?

Dandruff: You are asking two questions. Dandruff can be caused by dry skin, but it can be caused by other conditions like psoriasis. And yes, allergies can cause dry skin.

Eczema?: So lots of things can affect dryness or oiliness of skin. Allergies have been known to affect skin texture as well. One condition eczema which typically shows up in young kids often first on arms and legs. In adults typically effects more face and scalp.