I have been working in question 2 and I think I have a good explanation. At the end is a triviality, but that's what (almost) always happens in math when you understand something.

I am going to discuss here the Euclidean plane from two different perspectives.

# Euclidean plane
## From the point of view of classical differential geometry
The Euclidean plane is the manifold $M=\mathbb{R}^2$, with coordinates $(x_1,x_2)$, together with the natural [Riemannian metric][1] 
$$
g=dx_1\otimes dx_1+dx_2\otimes dx_2.
$$
It is, therefore, a Riemannian manifold.

Seen as a plain manifold we can consider that $M$ is endowed with a natural linear connection $\nabla$, such that $\nabla_{\partial_{x_i}}\partial_{x_j}=0\partial_{x_1}+0\partial_{x_2}$. The [induced][2] principal connection on the frame bundle is given by the 1-form
$$
\omega=
\begin{pmatrix}
c_{11}&c_{12}\\
c_{21}&c_{22}\\
\end{pmatrix}^{-1}\cdot\begin{pmatrix}
dc_{11}&dc_{12}\\
dc_{21}&dc_{22}\\
\end{pmatrix}\in \Omega(\mathbb R^2,\mathfrak{gl}(2))
$$
at 
$$
f=
\begin{pmatrix}
c_{11} & c_{12}& x_1\\
c_{21} & c_{22}& x_2\\
0 & 0& 1\\
\end{pmatrix}\in FM
$$
[Here][2] it is shown how to construct $\omega$ from an arbitrary $\nabla$.

On the other hand, if we think of $M$ as a Riemannian manifold we can consider the [Levi-Civita connection][3] $\nabla_{LC}$. Since the metric is constant, the covariant derivative $\nabla_{LC}$ coincides with the natural covariant derivative $\nabla$, so it induces the same connection $\omega$ on $FM$.
But the metric $g$ also specifies a orthonormal frame bundle $OM$ ([see here why][4]). The elements of this principal bundle are
$$
f=
\begin{pmatrix}
c & -\sqrt{1-c^2}& x_1\\
\sqrt{1-c^2} & c& x_2\\
0 & 0& 1\\
\end{pmatrix}\in OM
$$
with $c\in [-1,1]$. 
Since
$$
\omega|_{OM}=\begin{pmatrix}
c&\sqrt{1-c^2}\\
-\sqrt{1-c^2}&c\\
\end{pmatrix}\cdot\begin{pmatrix}
dc&\frac{c}{\sqrt{1-c^2}}dc\\
\frac{-c}{\sqrt{1-c^2}}dc&dc\\
\end{pmatrix}=
$$
$$
=\begin{pmatrix}
0&\frac{dc}{\sqrt{1-c^2}}\\
-\frac{dc}{\sqrt{1-c^2}}&0\\
\end{pmatrix} \in \Omega(\mathbb R^2,\mathfrak{o}(2)),
$$
according to Proposition 4.7. in [vicenteBundles](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwj-w6a2xYj7AhVJ3hoKHVDpAMQQFnoECAsQAQ&url=http%3A%2F%2Fwww.mat.ucm.es%2F~vmunozve%2FNotas-fibrados.pdf&usg=AOvVaw2EmcybsLZEb16epRLqS8z8), this connection can be reduced to a connection on the orthonormal frame bundle determined by the metric $g$. 

If we parameterize this principal bundle $OM$ with
$$
(x_1,x_2,\theta)\mapsto \begin{pmatrix}
\cos(\theta) & -\sin(\theta)& x_1\\
\sin(\theta) & \cos(\theta)& x_2\\
0 & 0& 1\\
\end{pmatrix}
$$
we obtain the more famous expression for $\omega$:
$$
\begin{pmatrix}
0&d\theta\\
-d\theta&0\\
\end{pmatrix}
$$
Remember: this 1-form **tells us how much the bases at $f$ and $f'$ "fail to be constant"** when we pass from the frame $f$ to a nearby frame $f'$, but expressing this mistake with respect to the frame $f$ itself.

## From the point of view of Cartan geometry

The Euclidean plane is a Cartan geometry modeled over $(E(2),O(2))$ , indeed **is** the Klein geometry $(E(2),O(2))$. Moreover, it is a [reductive Klein geometry][5] since
$$
\mathfrak e(2)=\left\{\begin{pmatrix}
C & v\\
0 & 0\\
\end{pmatrix} 
:C\in \mathfrak{o}(2), v\in \mathbb R^2\right\}=\mathfrak o(2)\oplus \mathfrak p
$$
We have a natural choice for $\mathfrak p$
$$
\mathfrak{p}=\left\{\begin{pmatrix}
0 & p\\
0 & 0\\
\end{pmatrix} 
:p\in \mathbb R^2\right\}.
$$

With this in mind, remember that the Maurer-Cartan form describes all possible "infinitesimal displacements" of the frame $f$, but from the point of view of the frame $f$ itself. That is, if we pass from the frame $f$ to another frame $f'$, the Maurer-Cartan form at $f$ applied to the "vector" $\vec{ff'}=(dx_1,dx_2,d\theta)$ is a packet of information
$$
A=\begin{pmatrix}
0&-d\theta&cos(\theta)dx_1+sin(\theta)dx_2\\
d\theta&0&-sin(\theta)dx_1+cos(\theta)dx_2\\
0&0&0&\\
\end{pmatrix}\in \mathfrak e(2)
$$
Here is encoded, on the one hand, how much have we moved the base point of $f$ to the base point of $f'$ and, on the other, how much have we changed the basis itself. The natural choice of $\mathfrak p$ let us think that the information about the change of base point is in the $v$ part (the projection of the Maurer-Cartan form on $\mathfrak p$), and so the projection of the Maurer-Cartan form on $\mathfrak o(2)$, $\begin{pmatrix}0&d\theta\\-d\theta&0\\\end{pmatrix}$, tell us how much has the basis changed. That is, the same as the connection 1-form of the connection on the orthonormal bundle induced by the metric $g$ (which is the Levi-Civita connection).

**To summarize:**  
In the orthonormal frame bundle induced by the metric $g$ we consider a displacement from a frame $f=\begin{pmatrix}C & p\\0 & 1\\\end{pmatrix}$ to a frame $f'=\begin{pmatrix}C' & p'\\0 & 1\\\end{pmatrix}$. 

The principal connection $\omega$ induced by the Levi-Civita connection measures the change from $C$ to $C'$ as an element of $\mathfrak o(2)$. 

The Cartan connection (Maurer-Cartan form) measures the change from $f$ to $f'$ as an element of $\mathfrak e(2)$. This change can be decomposed like the union of a change from $C$ to $C'$ and a change from $p$ to $p'$. This is reflected in the fact that  $\mathfrak e(2)=\mathfrak o(2)\oplus \mathfrak p$. If we focus on the change from $C$ to $C'$ we have the principal connection $\omega$.



  [1]:https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/GEOMETRY/Riemannian%20metric.md.html
  [2]:https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/GEOMETRY/associated%20connection.md.html#Example
  [3]:https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/GEOMETRY/Levi-Civita%20connection.md.html
  [4]:https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/orthonormal%20frame%20bundle.md.html
  [5]:https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/reductive%20Cartan%20geometry.md.html#Interpretation