I am given the definition of the Paneitz operator for Riemannian 4-manifolds as follows:
\begin{equation} P = (-\Delta)^2 + \delta\big(\frac{2}{3}Rg - 2\operatorname{Ric}\big)d \end{equation}
where $\Delta$ is the Laplace-Beltrami operator for our fixed metric $g$, $\delta$ is the negative divergence, $R$ is the scalar curvature, $\operatorname{Ric}$ is the Ricci curvature (all with respect to $g$) and $d$ is the differential (acting on functions).
I am struggling to make sense of this: obviously I understand how $(-\Delta)^2$ is an operator acting on functions, but the other part of the RHS I don't understand. If $d$ is to act on a function $f$ to produce a 1-form $df$, and $\delta$ being the negative divergence is to act on a vector field, how can
\begin{equation} (\frac{2}{3}Rg - 2\operatorname{Ric}\big) \, df \end{equation}
be interpreted as a vector field? This term doesn't make sense to me in any case - in the brackets we have a $(0,2)$-tensor, which even if I were to apply a type change to produce a $(1,1)$-tensor, after acting on $df$ this would still give me a $(0,1)$-tensor field rather than a vector field! Should I be considering $g$ and $\mathrm{Ric}$ as their $(2,0)$-tensor types instead? Rather, I imagine it is my understanding of divergence that is wrong here - could someone could please elucidate things? Thank you.
There are some implicit musical isomorphisms (i.e., index-raising and index-lowering operations) going on there that are not stated. First, although $df$ is a $1$-form, by means of the metric we can turn it into a vector field $(df)^\sharp = \operatorname{grad} f$.
Next, any covariant $2$-tensor field $T$ can be interpreted as a bundle map from vector fields to covector fields, by sending a vector field $X$ to the $1$-form $T(X,\cdot)$. Thus we can interpret $\big(\frac 2 3 R\, g - 2\operatorname{Ric}\big)df$ as the $1$-form $$ \big(\tfrac 2 3 R\, g - 2\operatorname{Ric}\big)(\operatorname{grad} f, \cdot). $$
Finally, apply one more musical isomorphism to turn this $1$-form into a vector field, and then apply the divergence to it.
People who work a lot with Riemannian geometry often get into the habit of just raising and lowering indices automatically without saying anything about it.