A) Given a non-constant polynomial $q\in\mathbb{Z}[\alpha_1,\alpha_2,\ldots,\alpha_n],$ if we pick random $\omega_i\in\mathbb{F}$ (a finite field) uniformly and independently across $1\leq i\leq n,$ then, we know that $q(\omega_1,\omega_2,\ldots,\omega_n)\neq 0$ with high probability (i.e. the probability goes to 1 as $|\mathbb{F}|\rightarrow\infty$).

B) Given another polynomial $r\in\mathbb{Z}[\alpha_1,\alpha_2,\ldots,\alpha_n],$ I am interested in determining if there exists a field $\mathbb{F}$ and a choice of $\omega_i\in\mathbb{F}$ which simultaneously satisfy $q(\omega_1,\omega_2,\ldots,\omega_n)\neq 0$ and $r(\omega_1,\omega_2,\ldots,\omega_n)= 0.$ Is there a theorem that gives necessary or sufficient conditions for this to happen? Is it true that if it happens over some field, then it happens over all sufficiently large finite fields?

Is it true that if there is a point which satisfies $r=0$ and $q\neq 0,$ then "most" of the points satisfying $r=0$ also satisfy $q\neq 0,$ in similar spirit to the result A which is the case of $r$ being the zero polynomial?

I am interested only in solutions over finite fields and not over their algebraic closures.

Thanks a lot.