I will quibble with your listing the Fano Plane under graph theory rather than under geometry or perhaps combinatorics. Now I know people who do not think the theory of finite planes is very geometric, and I agree there is much truth here. In some cases it requires algebraic rather than geometrical work to make progress, and in other cases combinatorial ideas. Yet, trying to "imitate" in the finite plane world interesting geometric phenomenon in the Euclidean, projective, or hyperbolic planes I think has proved very fruitful.
I don't really see that the Fano Plane leads to graph theory questions that are of great interest, and that would not be raised from some other point of view.