In question 1, I think you are just asking, if we have elements of $G$ written as $$g=\begin{pmatrix}h&v\\ 0&1\end{pmatrix}$$ then $$g^{-1}=\begin{pmatrix}h^{-1}&-h^{-1}v\\ 0&1\end{pmatrix}$$ hence $$g^{-1}dg=\begin{pmatrix}h^{-1}dh&-h^{-1}dv\\ 0&1\end{pmatrix}.$$ This is clearly since the Lie group operation is matrix multiplication.
In books which discuss the moving frame, one learns that the soldering forms $\omega_1,\omega_2$ are precisely those for which the projection of orthonormal frame bundle to surface pulls back the dual of the orthonormal frame to $\omega_1,\omega_2$ and the Levi--Civita connection form $\omega_{12}$ is precisely the one for which $d\omega_1=-\omega_{12}\wedge\omega_2$ and $d\omega_2=\omega_{12}\wedge\omega_1$, so you can check that $\mu_3$ is the Levi--Civita. For an example of such a book, my Introduction to Exterior Differential Systems discusses all of this in its many appendices, in particular in appendix G the structure equations are derived for Riemannian metrics on surfaces.