For the first question, if X has codimension 1, it means that you can fix an element and then any element y in A can be uniquely expressed in the form , where and is a scalar. Define . Then f is a linear functional whose zero set is X.
For the second question, if then (since f(e)=1). Thus y is in the zero set of f, which by hypothesis consists of non-invertible elements. But if an element is non-invertible then 0 is in its spectrum.