Some questions about Weyl spinor algebra

I am following a review on supersymmetry and I am having trouble with Weyl spinor algebra. These are equations (4.2.1)

$$Q_{\alpha}=i\frac{\partial}{\partial\theta^{\alpha}}-(\sigma^{\mu}\theta^{\dagger})_{\alpha}\partial_{\mu}\qquad{}Q^{\alpha}=-i\frac{\partial}{\partial\theta_{\alpha}}+(\theta^{\dagger}\bar{\sigma}^{\mu})^{\alpha}\partial_{\mu}$$

in order to get some familiarity with spinor gimnastics I am trying to get one from the other raising the index

$$Q^{\alpha}=\epsilon^{\alpha\beta}Q_{\beta}=$$ $$=i\epsilon^{\alpha\beta}\frac{\partial}{\partial\theta^{\beta}}-\epsilon^{\alpha\beta}((\sigma^{\mu})_{\beta\dot{\beta}}\theta^{\dagger\dot{\beta}})\partial_{\mu}$$ here my first question arises, why does raising the index of $\frac{\partial}{\partial\theta^{\beta}}$ give a minus sign? focusing now on the second term of the last equation

$$-\epsilon^{\alpha\beta}((\sigma^{\mu})_{\beta\dot{\beta}}\theta^{\dagger\dot{\beta}})\partial_{\mu}=-((\sigma^{\mu})^{\alpha}_{\,\,\,\dot{\beta}}\epsilon^{\dot{\beta}\dot{\gamma}}\theta^{\dagger}_{\,\,\,\dot{\gamma}})\partial_{\mu}=\epsilon^{\dot{\gamma}\dot{\beta}}((\sigma^{\mu})^{\alpha}_{\,\,\,\dot{\beta}}\theta^{\dagger}_{\,\,\,\dot{\gamma}})\partial_{\mu}=((\sigma^{\mu})^{\alpha\dot{\gamma}}\theta^{\dagger}_{\,\,\,\dot{\gamma}})\partial_{\mu}$$ this last equation should give $(\theta^{\dagger}\bar{\sigma}^{\mu})^{\alpha}\partial_{\mu}$ and here comes my second question, how does this follow?