This identifies $Der(A)$ with the subquotient $Der(R,I)/I$ of $Der(R)$, where $Der(R,I) < Der(R)$ is the module of derivations preserving $I$. Note that $Der(R,I)$ is the kernel of the natural $R$-module map $Der(R) \to Hom_R(I/I^2,A)$ defined by evaluating derivations on elements of the ideal. This makes it fairly straightforward to get Macaulay2 to calculate $Der(A)$ in terms of generators and relations as you suggest; it it probably equivalent to the algorithm you described.

I think every (multi)derivation of $A$ lifts to one on $R$. Let $y_i = x_i \mod I$ be the images of the generators. Then a derivation of $A$ is completely determined by how it acts on each $y_i$. Suppose that $Z : A \to A$ is a derivation and let $Z_i = Z(y_i) \in A$. Now choose lifts $\tilde Z_i \in R$ of these elements and set $\tilde Z = \sum_i \tilde Z_i \partial_{x_i}$. Then $\tilde Z$ is a derivation of $R$ that lifts $Z$. The case $p > 1$ is similar.

You might be interested in this paper of Manetti: numdam.org/item/ASNSP_2007_5_6_4_631_0

Nicola's answer below certainly gives the desired counterexample. I just wanted to mention that if $I$ is already radical then there is a simple obstruction to equality of $I$ and $\langle I\cap Z \rangle$. Namely, the Poisson bracket descends to a Lie algebra structure on $I/I^2$ (the "co-normal Lie algebra"). If $I$ is generated by elements of $Z$ then this Lie bracket will be abelian, but most conormal algebras are not. For example, for the maximal ideal $\mathfrak{m}$ defining the origin in the dual of a Lie algebra $\mathfrak{g}$, we have $\mathfrak{m}/\mathfrak{m}^2\cong\mathfrak{g}$

You're welcome, @Eric; I hope it helps!

Thanks for the references, @StevenSam. Those papers were, indeed, part of the motivation for the question. In fact, the proof that the Poisson divisors I mentioned have codimension three Gorenstein singular loci proceeds by expressing the singular locus as the degeneracy locus of a certain skew form obtained from the Poisson structure.