Why can a gene lack of a binding site be expressed in skin cells?
You understand this wrong: All present enhancers of a gene must be bound by a enhancing factor, not all factors of a certain cell types have to be bound. So when skin cells contain activators for B, D and E they can activate gene 2 and 4 of your example, but not gene 3 since the activators for A and C are missing.
So when you have a gene containing any combination of B, D and E, its activation in skin cells is possible according to this model. The same is true for any combination of A, C and E for liver cells. The special case would be a gene having E as the only enhancer as this could be activated in both, liver and skin cells.