The answer to your second question is no. Let $X$ be a three-dimensional algebraic torus, then the automorphism group of $X$ is the extension of $GL_3(\mathbb Z)$ by $X$ (acting by translations. Let $U$ be the complement of three points in general positionin a one-dimensional subtorus. Then the automorphism group of $U$ is the subgroup of $GL_3(\mathbb Z)$ fixing a vector. If there were a surjective homomorphism $Aut(X) \to Aut(\mathbb Z)$, the image of the torus would be a normal abelian subgroup, hence when sent to $GL_2(\mathbb Z)$ it would lie in $\pm 1$, but there is no surjective homomorphism $GL_3(\mathbb Z) \to PGL_2(\mathbb Z)$.

The answer to your second question is no. Let $X$ be a three-dimensional algebraic torus, then the automorphism group of $X$ is the extension of $GL_3(\mathbb Z)$ by $X$ (acting by translations. Let $U$ be the complement of three points in general positionin a one-dimensional subtorus. Then the automorphism group of $U$ is the subgroup of $GL_3(\mathbb Z)$ fixing a vector. If there were a surjective homomorphism $Aut(X) \to Aut(U)$, the image of the torus would be a normal abelian subgroup, hence when sent to $GL_2(\mathbb Z)$ it would lie in $\pm 1$, but there is no surjective homomorphism $GL_3(\mathbb Z) \to PGL_2(\mathbb Z)$.