From my understanding, gauge fields are essentially physics terminology (probably originating from electrodynamics) for connections in some bundle of interest (here a vector bundle over $M$).
Next, note that $C^{\infty}(M,E)$ typically denotes all smooth maps from $M$ into $E$, so I find it odd that the paper uses it to denote the space of sections. If we wish to speak of only the smooth sections, the correct notation is something like $\Gamma(E)$ or $\Omega^0(M,E)$ ($E$-valued $0$-forms on $M$). The object $d_{A}$ is known as the exterior covariant derivative (which is typically denoted as $d_{\nabla}$ or $d^{\nabla}$). Also, yes, for a smooth section $\psi\in \Omega^0(M,E)$, we define $d_{\nabla}\psi:= \nabla\psi$. The reason it becomes an $E$-valued $1$-form on $M$ is because $\nabla_{(\cdot)}\psi$ has a slot which can be
filled in by an element of $TM$, and if you feed it a $\xi_x\in T_xM$ then the output is $\nabla_{\xi_x}\psi\in E_x$.
Finally, regarding Hodge stars, let us first consider the case of vector spaces; we can then generalize to vector bundles by doing things fiber-by-fiber. Let $(V,\langle\cdot,\cdot\rangle)$ be an $n$-dimensional oriented pseudo-inner product space
and let $E$ be a vector space. Note that an $E$-valued $k$-form on $V$ can be defined in several equivalent ways:
- an alternating multiliner map $\underbrace{V\times\cdots \times V}_{\text{$k$ times}}\to E$
- a linear map $\bigwedge^k(V)\to E$
- an element of $\left(\bigwedge^k(V)\right)^*\otimes E\cong \left(\bigwedge^k(V^*)\right)\otimes E$
We shall use the third description. Let $\star:\bigwedge^k(V^*)\to \bigwedge^{n-k}(V^*)$ be the usual Hodge-star defined by the orientation and pseudo-inner product of $V$. This allows us to consider a bilinear map $\bigwedge^k(V^*)\times E\to \bigwedge^{n-k}(V^*)\otimes E$ defined as $(\alpha,\xi)\mapsto (\star\alpha)\otimes \xi$. Since this is a bilinear map, the universal property of tensor products tells us that we get a corresponding linear map $\star_E:\bigwedge^k(V^*)\otimes E\to \bigwedge^{n-k}(V^*)\otimes E$ such that for all "pure tensors" $\alpha\otimes \xi\in \bigwedge^{k}(V^*)\otimes E$, we have
\begin{align}
\star_E(\alpha\otimes \xi)&=(\star \alpha)\otimes \xi.
\end{align}
Hence, $\star_E$ is the desired mapping. It takes $E$-valued $k$-forms on $V$ to $E$-valued $(n-k)$-forms on $V$, and this definition is such that in the special case $E=\Bbb{R}$ is the underlying field, it coincides with the usual definition of $\star$ (after identifying a vector space of the form $W\otimes \Bbb{R}$ with $W$ in the natural way).
Now that we know how Hodge-star works at the level of vector spaces, you can do it at the vector-bundle level by doing things fiber-by-fiber to get a vector bundle morphism $\star_E:\bigwedge^k(T^*M)\otimes E\to \bigwedge^{n-k}(T^*M)\otimes E$, and hence by applying this to forms pointwise, we obtain a mapping $\Omega^k(M,E)\to \Omega^{n-k}(M,E)$ for each value of $k$. In particular, if $\psi$ is a smooth section, then $d_{\nabla}\psi$ is an $E$-valued 1-form on $M$, so its Hodge star is an $E$-valued $(n-1)$-form on $M$.
