intersection of hyperplane with a set of finite points in $\mathbb{P}^3$ which fails to impose independent conditions for sections of $\mathcal{O}(d)$

Let $ X$ be a finite set of points in $ \mathbb{P}^3$ of cardinality $ \ge 3d +3$ which fails to impose independent conditions on sections of $ \mathcal{O}(d)$ and $ X$ does not pass through any quadratic hypersurface. Can we give an effective bound on cardinality of a subset $ Z \subset X$ such that there exist a hyper plane containing $ Z$ ?

If $ |X| < 3d + 2$ , then it is known that either there exist a line containing at least $ d+2$ points of $ X$ or there is a plane containing at least $ 2d+2$ points of $ X$ .

restricting sheaves in $\mathbb{P}^3$

Is it true that one has an exact sequence of the following form ? $ 0 \to \mathcal{O}_Z \to I_{Z, \mathbb{P}^3}(1)\otimes \mathcal{O}_H \to I_{Z, H}(1) \to 0$ , where $ Z$ is a finite set of points in $ \mathbb{P}^3$ contained in a hyperplane $ H$ and $ I_{Z, \mathbb{P}^3}$ and $ I_{Z, H}$ denote the ideal sheaf of $ Z$ in $ \mathbb{P}^3$ and $ H$ respectively