The group of orientation-preserving $H^\infty$-diffeomorphisms on $\mathbb{R}^n$ is defined as
$$\operatorname{Diff}(\mathbb{R}^n):=\{\operatorname{id}+f\,|\,f\in H^\infty(\mathbb{R}^n,\mathbb{R}^n)\text{ and } \det(\operatorname{id}+df)>0 \}$$ where $$H^\infty(\mathbb{R}^n,\mathbb{R}^n)=\bigcap_{q\geq0}H^q(\mathbb{R}^n,\mathbb{R}^n).$$ is the intersection of all Sobolev spaces $H^q(\mathbb{R}^n,\mathbb{R}^n)$.
I need an explanation for and/or motivation for this definition. To me, elements of $\operatorname{Diff}(\mathbb{R}^n)$ are just smooth (in the weak sense) functions that decay to the identity at infinity. However, elements of $\operatorname{Diff}(\mathbb{R}^n)$ ought to be invertible with smooth inverse.
$\textbf{ 1. Why are functions of the form $\operatorname{id}+f$ with $f\in H^\infty(\mathbb{R}^n,\mathbb{R}^n)$ invertible and what is the inverse?}$ I've looked here: Inverse of a sum of functions, and it doesn't appear there is a general rule for the inverse of a sum of functions.
$\textbf{2. Why define elements of $\operatorname{Diff}(\mathbb{R}^n)$ as shifts of the identity to begin with? Why not just define}$
$$\operatorname{Diff}(\mathbb{R}^n):=\{f\in H^\infty(\mathbb{R}^n,\mathbb{R}^n)\,|\,f^{-1}\in H^\infty(\mathbb{R}^n,\mathbb{R}^n)\text{ and } \det(\operatorname{id}+df)>0 \}?$$
This definition is given in this paper, at the beginning of section 2, or in this paper, under heading 3.4.