Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where $\delta_d:= \frac{d(d+3)}{2} $. Given a point $p \in \mathbb{P}^2$ define $H_p \subset \mathcal{D}$ to be the hyperplane $$ H_p := \{ [f] \in \mathcal{D}: f(p) =0 \}. $$ Suppose $\mathcal{V}$ and $\mathcal{W}$ are two smooth algebraic varieties inside $\mathcal{D}$ of dimension $k$ and $k-1$ respectively. I have three questions:

1) Suppose that for a "generic" choice of $p \in \mathbb{P}^2$, $H_p$ intersects both $\mathcal{V}$ and $\mathcal{W}$ transversally (ie the set of $p$ for which the above statement is false is contained inside a finite union of subvarieties of $\mathbb{P}^2$ of strictly smaller dimension.)

Is the following statement necessarily true: For generic choices of $k$ points $p_1, \ldots, p_k$ the intersection $$ \mathcal{V} \cap H_{p_1} \cap \ldots \cap H_{p_k} $$ is transverse and $$ \mathcal{W} \cap H_{p_1} \cap \ldots \cap H_{p_k} = \emptyset $$ ?

2) Same hypothesis as in 1). Is it true that for generic choice of points $p_1$ and $p_2$, the intersection of $H_{p_2}$ with $\mathcal{V} \cap H_{p_1}$ is transverse? Note that by assumption all the three pairwise intersections are transverse if $p_1$ and $p_2$ are generic: $H_{p_1}$ and $\mathcal{V} $; $H_{p_2}$ and $\mathcal{V} $; $H_{p_1}$ and $H_{p_2} $. But that doesn't mean that $H_{p_2}$ will intersect transversally with $\mathcal{V} \cap H_{p_1} $. The claim/hope is that I can delete a further "small" set from $\mathbb{P}^2 \times \mathbb{P}^2 $ to ensure that the desired triple intersection is transverse.

3) Let us make a much stronger assumption: For generic choices of $k$ points $p_1 \ldots p_k$ the intersection $$ \mathcal{V} \cap H_{p_1} \cap \ldots \cap H_{p_k} $$ is transverse. Does it imply for generic choices of $k+1$ points, the following intersection is empty, ie $$ \mathcal{V} \cap H_{p_1} \cap \ldots \cap H_{p_{k+1}} = \emptyset $$ ? Note that any $k+1$ fold intersection in the above expression is transverse (ie the intersection will be a finite bunch of points). That does not mean the $k+2$ fold intersection will be transverse (hence empty). But does it mean for generic choices of $k+1$ points the $k+2$ fold intersection will be empty? Forget about $\mathcal{W}$ for this question.