Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\text{ad}:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let $\text{ad}P:=P\times_{\text{ad}} \mathfrak{g}$ be the adjoint bundle.
I am seeking a bit of clarification on the sections of $\text{ad}P$.
Let $\sigma\in \Gamma(\text{ad}P)$. Since $\sigma$ is a map $M\rightarrow \text{ad}P$, it follows that $\sigma(m)\in \text{ad}P$ and so
$\sigma(m)=[(p,v)]$ for some $p\in P$ and $v\in \mathfrak{g}$.
(Here $[,]$ denotes the equivalence class defined by the right action of $G$).
However, since $\sigma$ is a section of the vector bundle $\pi_{\mathfrak{g}}:\text{ad}P\rightarrow M$, it follows that $\sigma(m)\in \pi_{\mathfrak{g}}^{-1}(m)$.
The space $\pi_{\mathfrak{g}}^{-1}(m)$ has the structure of a vector space. Thus, $[(p,v)]$ is an element of a vector space.
Question 1: is it true to say that
\begin{align*} \pi_{\mathfrak{g}}^{-1}(m)=\{[(p,v)]:\pi(p)=m,v\in \mathfrak{g}\} \end{align*}
Question 2: how are the vector space operations defined? Here is what I think is a natural way but I have no idea if I'm correct.
If $p,q\in\pi^{-1}(m)$, and $v,w\in \mathfrak{g}$, then how do we define
$[(p,v)]+[(q,w)]$ ?
Is it by noting that since $p$ and $g$ lie in the same fibre over $m$, then $q=pg$ for some $g\in G$, and thus
\begin{align*} [(q,w)]=[(pg,w)]=[(p,\text{ad}_g w)g]=[(p,\text{ad}_g w)]. \end{align*}
And so, \begin{align*} [(p,v)]+[(q,w)]=[(p,v+\text{ad}_g w)] \end{align*} for $g\in G$ such that $q=pg$?
What you have written looks correct to me. A nice way to look at sections of any associated bundle is as equivariant smooth functions on the total space of the principal bundle. In the case you are looking at, this means to a section $\sigma$ of ad$P$ can be equivalently encoded as a smooth function $f:P\to\mathfrak g$ such that $f(p\cdot g)=\operatorname{ad}_{g^{-1}}(f(p))$. The correspondence is characterized by $\sigma(m)=[(p,f(p))]$ for each $p$ which lies over $M$. (Given $f$, you observe that you obtain the same class for each $p$, so $\sigma$ is well defined. Conversely, given $p$ over $M$, any point of ad$P$ lying over $m$ can be uniquely written as $[(p,X)]$ for $X\in\mathfrak g$, which allows you to construct $f$ from $\sigma$.) The nice point about this is that now the vector bundle structure is just given by point-wise addition of the corresponding functions.
This works for any associated bundle, and there also is a similar description of bundle valued differential forms.