Unfortunately, now I cannot devote much time to your question, thus the answer below can be superficial and not very checked, so I shall try to improve it later. On the other hand, I expect it should provide a positive answer to both your questions.
Let $V=\{(v_1,\dots,v_n)\in\{0,1\}^n:\sum_{i=1}^n d_iv_i\le s\}$ be the set of all admissible "bin packings". Then the convex hull $\operatorname{conv} V$ of $V$ is the set of all possible ``schedules''. Let $e=(1,1,\dots,1)\in\mathbb R^n$. Then $r_\max=\sup\{r\in [0,1]:re\in\operatorname{conv} V\}$. Since the set $\operatorname{conv} V$ is compact, $r_\max$ is attained, that is $r_\max e\in \operatorname{conv} V$. By Carathéodory's theorem, there exists an $n$ dimensional simplex $S$ with vertices in $V$, such that $r_\max e\in S$. Clearly, that $r_\max e$ is a boundary point of $S$. Let $V'$ be the set of the vertices of the face of $S$ of minimal dimension containing $r_\max e$. Then $|V'|\le n$ and there exists $(\lambda_v)_{v\in V'}$, such that $\lambda_v>0$ for each $v\in V'$, $\sum_{v\in V'}\lambda_v=1$, and $r_\max e =\sum_{v\in V'}\lambda_v v$. Put $f=\sum_{v\in V'}\lambda_v (nv-(v,e)e)$. It is easy to check that $f$ is both collinear and orthogonal to $e$, so $f=0$. Thus the family $(nv-(v,e)e)_{v\in V'}$ of vectors is linearly dependent. I expect that Lemma from this answer implies that there exist integers $(\lambda_f)_{v\in V'}$ bounded by a function of $n$, which are not all zeroes and $\sum_{v\in V'}f_v (nv-(v,e)e)$. I expect that the minimality of $V'$ implies that $(f_v)_{v\in V'}$ is proportional to $(\lambda_v)_{v\in V'}$, so $(f_v)_{v\in V'}$ provides the required some-times-bin-packing.