Let $I = (0, 1)$. I have two questions.
Let $p > 1$. For all $\epsilon > 0$, does there necessarily exist $C = C(\epsilon, m, p)$ such that$$\sum_{j = 0}^{m-1} \|D^j u\|_{L^\infty(I)} \le \epsilon\|D^m u\|_{L^p(I)} + C\|u\|_{L^1(I)} \text{ for all }u \in W^{m, p}(I)?$$
Let $1 \le q < \infty$. Do we necessarily have that for all $\epsilon > 0$, there exists $C = (\epsilon, q)$ such that$$\|D^{(m-1)}u\|_{L^q(I)} + \sum_{j = 0}^{m-2} \|D^j u\|_{L^\infty(I)} \le \epsilon\|D^m u\|_{L^1(I)} + C\|u\|_{L^1(I)} \text{ for all }u \in W^{m, 1}(I)?$$
For the first question, the answer is yes.
Proof: Take $\varepsilon>0$. Let $c>0$ be a constant such that $\|\cdot\|_{L^p}\leq c\|\cdot\|_{L^\infty}$. Set $\eta=\frac{\varepsilon}{1+c\varepsilon}$.
Since $W^{m,p}(I)\subset C^{m-1}(\overline{I})$ is compact and $C^{m-1}(\overline{I})\subset L^1(I)$ is continuous, the lemma in this post implies that there exists a constant $C_\eta>0$ such that \begin{align} \sum_{j=0}^{m-1}\|D^ju\|_{L^\infty}&=\|u\|_{C^{m-1}}\\ &\leq\eta\|u\|_{W^{m,p}}+C_\eta\|u\|_{L^1}\\\\ &=\eta \sum_{j=0}^{m-1}\|D^ju\|_{L^p}+\eta\|D^mu\|_{L^p}+C_\eta\|u\|_{L^1}\\\\ &\leq c\eta \sum_{j=0}^{m-1}\|D^ju\|_{L^\infty}+\eta\|D^mu\|_{L^p}+C_\eta\|u\|_{L^1} \end{align} and thus $$(1-c\eta)\sum_{j=0}^{m-1}\|D^ju\|_{L^\infty}\leq\eta\|D^mu\|_{L^p}+C_\eta\|u\|_{L^1}$$ Taking $C=\frac{C_\eta}{1-c\eta}$, we get $$\sum_{j=0}^{m-1}\|D^ju\|_{L^\infty}\leq\frac{\eta}{1-c\eta}\|D^mu\|_{L^p}+\frac{C_\eta}{1-c\eta}\|u\|_{L^1}=\epsilon \|D^mu\|_{L^p}+C\|u\|_{L^1}$$ as desired.