(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.

(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) I don't know.:)

On the one 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. Next one can try to proceed using the Tannkian formalism.

On the other hand, one can possibly find zero divisors using an argument similar to Lemma 3 in Poonen's https://arxiv.org/pdf/math/0204306.pdf.

(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) 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. Does this help?

(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.