Automorphism groups of affine varieties and a characterization of affine $n$-space

We show that the automorphism group of affine $n$-space $\mathbb{A}^n$ determines $\mathbb{A}^n$ up to isomorphism: If $X$ is a connected affine variety such that $\mathrm{Aut}(X) \simeq \mathrm{Aut}(\mathbb{A}^n)$ as ind-groups, then $X \simeq \mathbb{A}^n$ as varieties.
We also show that every torus appears as $\mathrm{Aut}(X)$ for a suitable irreducible affine variety $X$, but that $\mathrm{Aut}(X)$ cannot be isomorphic to a semisimple group. In fact, if $\mathrm{Aut}(X)$ is finite dimensional and if $X \not\simeq \mathbb{A}^1$, then the connected component $\mathrm{Aut}(X)^{\circ}$ is a torus.
Concerning the structure of $\mathrm{Aut}(\mathbb{A}^n)$ we prove that any homomorphism $\mathrm{Aut}(\mathbb{A}^n) \to \mathcal{G}$ of ind-groups either factors through $\mathrm{jac}\colon{\mathrm{Aut}(\mathbb{A}^n)} \to {\Bbbk^*}$ where $\mathrm{jac}$ is the Jacobian determinant, or it is a closed immersion. For $\mathrm{SAut}(\mathbb{A}^n):=\ker(\mathrm{jac})\subset \mathrm{Aut}(\mathbb{A}^n)$ we show that every nontrivial homomorphism $\mathrm{SAut}(\mathbb{A}^n) \to \mathcal{G}$ is a closed immersion.
Finally, we prove that every non-trivial homomorphism $\phi\colon{\mathrm{SAut}(\mathbb{A}^n)} \to\mathrm{SAut}(\mathbb{A}^n)$ is an automorphism, and that $\phi$ is given by conjugation with an element from $\mathrm{Aut}(\mathbb{A}^n)$.