Let $M^3$ denote the solid bitorus, and denote by $D$ the disk "spanned" by the curve in the picture. Is it correct to say that $[D] = 0$ in $H_2(M, \partial M; \mathbb{Z})$? I think the answer is yes, because $D$ separates $M$. Is this reasoning correct?
Yes, it is the boundary of the solid torus on either the left or the right, so is null homologous. To make this rigorous, you would triangulate this solid torus and then note that the boundary is composed of elements that lie in the boundary of the entire manifold (which are zero in the relative homology group) and the disk you mention.