Absolutely not. Take the the \ell_1$\ell_1$-sum of \ell_\infty$\ell_\infty^n$ (n$n\in \mathbb N$).
Let me answer the second question later If that embedded into $L_1(\mu)$, then you would have found $c_0$ in some ultrapower of $L_1(\mu)$, which is impossible.