English for "prolongement" or "Fortsetzung"?

Both "extension" and "continuation" are OK. I would be inclined to reserve "continuation" for the analytic continuation of analytic functions (or similar cases where the extension is somehow determined by its restriction to the original domain).

Yes, if $X\subset Y$ and $f:X\to Z$ and $g: Y\to Z$ such that $f(x)=g(x)$ for all $x\in X$, it is possible to call $g$ an extension of $f$, or to just say that "$g$ extends $f$."

"Continuation" connotes that the extension is somehow generated by the function and the subset, but in most cases a general function could be defined quite arbitrarily outside of $X$, so it does not seem to be a good substitute for "extension."

If you'd like to know the counterpart to the phrase "$g$ extends $f$," then we would speak of "restriction." That is, "$f$ is equal to the restriction of $g$ to the set $X$."