I am confused about a basic question. Consider the field equation of a gauge theory written in terms of curvilinear coordinates $$\frac{1}{\sqrt{-g}}D_\mu(\sqrt{-g}F^{\mu\nu})=j^\nu.$$ My questions are:

1. Do the partial derivatives in the gauge covariant derivative and the field strength remain partial, that is $D_\mu=\partial_\mu+ieA_\mu$ and $F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$, or they became covariant, $D_\mu=\nabla_\mu+ieA_\mu$ and $F^{\mu\nu}=\nabla^\mu A^\nu-\nabla^\nu A^\mu$ ?
2. I would like to lower all indices. Why should I write $$\frac{1}{\sqrt{-g}}D_\mu(\sqrt{-g}g^{\mu\sigma}g^{\nu\rho}F_{\sigma\rho})$$ instead of $$\frac{1}{\sqrt{-g}}g^{\mu\sigma}g^{\nu\rho}D_\mu(\sqrt{-g}F_{\sigma\rho}),$$ when the metric is position dependent? If the partial derivatives become covariant, then the metric can go in and out freely but I think that the derivatives are partial.