In answer to the first question:

The zero sets of $I$ and $J$ are not enough to know whether the generators of $I$ remain a regular sequence in $\mathbb{C}[u]/J$. In particular, the zero set of $J$ is not sensitive to changes in $J$ that do not affect its radical, but these changes can be relevant to whether the generators of $I$ remain a regular sequence.

This is shown by the example Jason Starr gave in answer to [your previous question][1]. For simplicity, consider the case $m=2$, $k=1$, i.e. $\mathbb{C}[u] = \mathbb{C}[x,y]$, and let $I=(y)$, $J=(x^2,xy)$ (per Jason's example). Then the zero set of $J$ is $\{x=0\}$, and the zero set of $I$ is $\{y=0\}$. In this situation, the generator $y$ of $I$ does not remain regular in $\mathbb{C}[x,y]/J$, since it becomes a zerodivisor.

On the other hand, if $J=(x)$ (note this is the radical of the previous choice of $J$, thus it has the same zero set), then $y$ remains regular in $\mathbb{C}[x,y]/J$.

Jason answered the second question in comments: a regular sequence in $\mathbb{C}[u]/J$ remains regular when you invert $u$, as it does in all flat extensions.

  [1]: https://mathoverflow.net/questions/311069/on-finite-dimensional-commutative-algebras-and-regular-sequences