(1) As pointed out by Marc Hoyois, there is a natural homomorphism $K_0(Var)\to K_0(DM^{gm})$, where $DM^{gm}$ is the category of geometric Voevodsky motives with coefficients in any (commutative unital) ring $R$; you should only assume (at our current level of knowledge on the resolution of singularities) that the base field characteristic $p$ is invertible in $R$ (to put Borel-Moore motives into $DM^{gm}$). Next, the Voevodsky embedding $Chow\to DM^{gm}$ induces an isomorphism on (the corresponding versions of) their $K_0$-groups; see Proposition 2.3.3 of my "$\mathbb{Z}[1/p]$-motivic resolution of singularities".
(3) This is not a domain in characteristic $0$ by the easy Lemma 3 in Poonen's https://arxiv.org/pdf/math/0204306.pdf (note that abelian varieties yield Chow motives if one considers them as complexes of sheaves with transfers, i.e., inside the Voevodsky category). I suspect that a similar argument can be applied in arbitrary characteristic.
On the other hand, one can easily prove that the "(more or less) standard" motivic conjectures predict that $K_0(Chow)$ is the free abelian group generated by isomorphism classes of indecomposable numerical motives.