Suppose that $f\colon X\to Y$ is a flat morphism of varieties over an algebraically closed field $k$. Let $E\subseteq X$ and $F\subseteq Y$ be closed subvarieties such that $f(E) = F$. Is it true that the restricted morphism $f|_E\colon E\to F$ is also flat? If not, are there some additional conditions on $f$ which would make this true?
No, the restriction $f:E\to F$ needn't be flat.
Take $Y=\mathbb A^2, X=\mathbb A^2 \times \mathbb P^1$ and for $f:X\to Y $ take the first projection, which is flat.
As to your second question, I am pessimistic about a general criterion ensuring that the restriction of a flat map will remain flat.