On page 352 of Hatcher's online text located here: https://www.math.cornell.edu/~hatcher/AT/ATpage.html , with the relevant chapter here: https://www.math.cornell.edu/~hatcher/AT/ATch4.pdf , he proves the CW approximation theorem. He does this by induction, in two steps. I have had great difficulty in understanding the description of step two.
What I think I understand: the k cells are attached by the constant map at the base point. But since he is attaching k-cells to A, shouldn't the resulting spheres be (k-1) spheres? I guess I don't really understand what he is saying in condition two.
I'm not sure I follow the part of your question asking whether the attachment should result in $(k-1)$-spheres.
When we attach a $k$-cell (via its boundary) to a point it's really just the same as identifying the boundary $S^{k-1}$ of $D^k$ to a point, so we end up attaching a copy of $D^k/S^{k-1}$ at a point. Then $D^k/S^{k-1}=S^k$, so we are indeed attaching a $k$-sphere as is claimed in the book.