I am not clear about what you find confusing. The answer to the first question seems simply to be that the expression $g^{-1}dg$ is meaningless on an abstract Lie group. To be more precise, the expression $g$ is subjected to the exterior derivative $d$ by first being interpreted as a function on the group $g \colon G \to GL(n)$, i.e. the inclusion function, if $G\subseteq GL(n)$. But on an abstract Lie group, there is no such inclusion function. So we need to find a property of $g^{-1}dg$ that holds on matrix Lie groups $G\subseteq GL(n)$, and which can generalize to all Lie groups.
I think Deane answered the other questions. For more detail, the definition of a wedge product of 1-forms $\alpha,\beta$ valued in an arbitrary algebra $A$ (not necessarily a Lie algebra) with multiplication $x\in A, y \in A \mapsto m(x,y)\in A$ is $m(\alpha,\beta)(v,w)=m(\alpha(v),\beta(w))-m(\alpha(w),\beta(v))$. I don't think there is any more reasonable definition. When applied to the Lie algebra of matrices under bracket, this yields $[\alpha,\beta]=2\alpha\wedge\beta$, if you expand it out. Hence the factor of 2 to relate Lie bracket to matrix multiplication.
We need the Lie bracket because, as Deane says, there is no other obvious product in the Lie algebra.