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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.