The identity in Question 1 is a special case of Theorem 2 in Ohno: A Generalization of the Duality and Sum Formulas on the Multiple Zeta Values (see [here][1]). Indeed, putting $k=1$ and $n=m$ in this theorem, and noting that $\zeta_1(s)$ equals $s\zeta(s+1)$, the result follows.

  [1]: http://www.sciencedirect.com/science/article/pii/S0022314X98923149