Transformation group in moving frame method

Question

I am reading the following paper: https://ieeexplore.ieee.org/document/4700863/ (just at the beginning of Section II-B)

My questions are the following:

1. What is $\partial_g\phi_g$ ? (partial differentiation of a diffeomorphism $\phi_g$ w.r.t one group element $g$? )
2. Why did that paper choose $(e,\xi^0)$, can we choose $(g,\xi^0)$ with $g\in G$?
3. Can $\phi_g^a$ be viewed as a matrix? (since the paper says $\phi_g^a$ is invertible).
4. If 3. is yes, then the paper also says $(\phi^a_g,\phi^b_g)$ has $r$ and $n-r$ components, what do components here mean? different sub-block of a big matrix?

I think 2. is related to tangent space of a diffeomorphism; however, I still want to get a more complete information.

Thanks!