First, the embedding problem has a partial affirmative answer if $k\geq n-2$ or if $n\geq 2k+2$. In these cases, $I$ is generated by $n-k$ elements, not necessarily variables.

Problem 1 has an affirmative answer. $I$ is generated by $y_j-f_{j-n}(y_1,\ldots, y_n)$ for $j>n$ and these are coordinates of $B$.