Freivalds matrix multiplication with non binary random vector
The answer to both questions is yes. The matrix $A B - C$ gives rise to a linear application over your field $\mathbb F$, which is a multivariate polynomial of total degree $d=1$. By the Schwartz-Zippel lemma, when $x$ is drawn from $S^n$ (where $S \subseteq {\mathbb F}$), the probability that $(A B - C) x = 0$ is at most $d/|S| = 1/|S|$, provided that $A B \neq C$. 
As you observe, this can also be proved directly using the standard argument for Freivald's algorithm.