Recall that for a line bundle $L$, we can take local trivialization $\phi:\pi^{-1}(U_\alpha)\to U_\alpha\times\mathbb C$. Take local frame (one section, since line bundle is of rank 1), say $s_\alpha=\phi^{-1}(\pi(\cdot),1)$, then any section of $L$ can be represented by $s|_{U_\alpha}=f_\alpha s_\alpha$, for some complex function $f_\alpha$. The transition functions are defined on $U_{\alpha\beta}:=U_\alpha\cap U_\beta$, as
$$
s_\alpha=g_{\alpha\beta}s_\beta.
$$
Thus, for the coordinates of $s$:
$$
f_\beta=g_{\alpha\beta}f_\alpha.
$$
Locally, a connection of $L$ can be writen as:
$$
\nabla=d+A_\alpha,
$$
where $A_\alpha$ are complex valued 1-form over $U_\alpha$.
In fact,
$$
[\nabla s]|_{U_\alpha}=\nabla(f_\alpha s_\alpha)=df_\alpha s_\alpha+f_\alpha\nabla s_\alpha,
$$
thus, if we define $\nabla s_\alpha:=A_\alpha s_\alpha$, then we get
$$
[\nabla s]|_{U_\alpha}=(df_\alpha+f_\alpha A_\alpha) s_\alpha,
$$
which is the precise mean when we write $\nabla=d+A_\alpha$.
We need to know the transition relation of $A_\alpha$. This can be compute as follows:
$$
A_\alpha g_{\alpha\beta}s_\beta=A_\alpha s_\alpha=\nabla s_\alpha=\nabla(g_{\alpha\beta} s_\beta)=dg_{\alpha\beta} s_\beta+g_{\alpha\beta}A_\beta s_\beta.
$$
Thus,
$$
A_\alpha g_{\alpha\beta}=dg_{\alpha\beta}+g_{\alpha\beta}A_\beta\implies
A_\beta=g_{\alpha\beta}^{-1}A_\alpha g_{\alpha\beta}-g_{\alpha\beta}^{-1}dg_{\alpha\beta}=A_\alpha-g_{\alpha\beta}^{-1}dg_{\alpha\beta}.
$$
To move on, I will use a equivalent definition of curvature: $F_\nabla=D^2$, where $D$ is the exterior differential. In particular, by the definition of $D$, we know that(locally)
\begin{align*}
D^2s&=D(\nabla s)=D[(d f_\alpha+f_\alpha A_\alpha)s_\alpha]\\
&=d(df_\alpha+A_\alpha f_\alpha)s_\alpha+(-1)^1(df_\alpha+A_\alpha f_\alpha)\wedge\nabla s_\alpha\\
&=[dA_\alpha f_\alpha-A_\alpha\wedge df_\alpha-(df_\alpha+A_\alpha f_\alpha)\wedge A_\alpha]s_\alpha\\
&=[dA_\alpha f_\alpha-A_\alpha\wedge df_\alpha+A_\alpha\wedge(df_\alpha+A_\alpha f_\alpha)]s_\alpha\\
&=[dA_\alpha+A_\alpha\wedge A_\alpha]f_\alpha s_\alpha\\
&=(dA_\alpha+A_\alpha\wedge A_\alpha)s.
\end{align*}
Therefore,
$$
F_\nabla|_{U_\alpha}=dA_\alpha+A_\alpha\wedge A_\alpha.
$$
Note that $A_\alpha$ is just a complex valued 1-form (in general it is a matrix valued 1-form), we have$A_\alpha\wedge A_\alpha=0$ and
$$
F_\nabla|_{U_\alpha}=dA_\alpha.
$$
Now, we are ready to show, $F_\nabla$ is a global defined 2-form. In fact,
\begin{align*}
F_\nabla|_{U_\beta}&=dA_\beta=d(A_\alpha-g_{\alpha\beta}^{-1}d g_{\alpha\beta})\\
&=dA_\alpha-dg_{\alpha\beta}^{-1}\wedge dg_{\alpha\beta}\\
&=dA_\alpha+g_{\alpha\beta}^{-1}dg_{\alpha\beta}g_{\alpha\beta}^{-1}\wedge dg_{\alpha\beta}\\
&=dA_\alpha+g_{\alpha\beta}^{-1}dg_{\alpha\beta}\wedge g_{\alpha\beta}^{-1}dg_{\alpha\beta}\\
&=dA_\alpha\\
&=F_\nabla|_{U_\alpha}.
\end{align*}
This answers your first question.
For the second, it is standard in Riemaniann geometry and I believe your can find in text book.
I would like to remark that, the above computation is trivial but maybe you need to do it once again by yourself.
Best Answer
Developing @Wyatt's second point ("The curvature is exactly how much the connection fails to be flat.") a bit further:
Curvature provides a useful tool to topologically classify vector bundles in terms of Chern-Weil theory.
Think of vector bundles over compact manifolds, for simplicity a line bundle over the torus, the latter thought as a square obtained as quotient of $\mathbb{R}^2$ by a lattice. For the vector bundle to be smooth, the fibers need to be periodic in a suitable sense and this claim restricts the freedom in choosing how they "twist over the torus". A good notion of a quantitative measure of "how they twist" is the curvature.
In fact, integrating the curvature over the torus gives you a topological invariant, that is, an integer number (up to some constant) which only depends on the topology of the bundle and is the same for whichever covariant derivative you have chosen. Moreover, if two vector bundles can be smoothly deformed into one another, then these numbers must be same, or conversely, if they are not the same, then the bundles cannot be smoothly deformed into one another. (Google for Chern numbers if you are interested). So "how much curvature there must be in total" is a topological property of the bundle, where by choosing different covariant derivatives, you can "distribute the curvature" in different ways over the manifold.
Going on from there, you can study Chern-Weil theory which provides a large class of such topological invariants by (up to some details) integrating suitable polynomials of the curvature (which become non-trivial if the base manifold has a higher dimension than 2). You can of course pretend to forget the notion of curvature and express these polynomials of the curvature as polynomials of the covariant derivative, but this makes it much harder to characterize these polynomials and would not be the usual way of mathematics in general to structurize theories as good as possible.