Faces of the intersection of convex sets

For finite-dimensional $V$, your definition of a face is equivalent to the definition of a poonem, according to part (i) of Exercise 7 on page 21 of the book Convex Polytopes by B. Gruenbaum; then the positive answer to your question is part (iii) of Exercise 9 on the same page.

It seems that the answer is Yes.

In the affine subspace $A$ spanned by $K:=K_1\cap K_2$, a face is the intersection of $K$ with a (closed) supporting hyperplane $\Pi$. By Hahn-Banach, there is an extension $\Pi_1$ of $\Pi$ as a closed hyperplane in $V$, so that $\Pi_1$ is a supporting hyperplane for $K_1$. Then $F_1=\Pi_1\cap K_1$ is a face of $K_1$. Likewise, $\Pi$ extends as a closed supporting hyperplane $\Pi_2$ of $K_2$, and $F_2=\Pi_2\cap K_2$ is a face of $K_2$. Eventually, $F=F_1\cap F_2$.