The answer to your first question is no.

In fact, consider the Grasmannian $G(2,4)$ of $2$-planes in $\mathbb{C}^4$, which has dimension $4$. It is isomorphic to the projective Grasmannian $\mathbb{G}(1,3)$ of lines in $\mathbb{P}^3$, and the Plucker embedding realizes it as a quadric hypersurface $X \subset \mathbb{P}^5$.

Now take a point $p$ and a plane $H$ in $\mathbb{P}^3$ such that $p \notin H$.

Let $\Sigma_p$ be the set of lines in $\mathbb{P}^3$ containing $p$ and let $\Sigma_H$ be the set of lines in $\mathbb{P}^3$ lying in $H$.

Then under the Plucker embedding $\Sigma_p$ and $\Sigma_H$ are carried to two linear subspaces of dimension $2$ in $X$ such that

$\Sigma_p \cap \Sigma_H= \emptyset$.

The answer to your first question is no.

In fact, consider the Grassmannian $G(2,4)$ of $2$-planes in $\mathbb{C}^4$, which has dimension $4$. It is isomorphic to the projective Grassmannian $\mathbb{G}(1,3)$ of lines in $\mathbb{P}^3$, and the Plücker embedding realizes it as a quadric hypersurface $X \subset \mathbb{P}^5$.

Now take a point $p$ and a plane $H$ in $\mathbb{P}^3$ such that $p \notin H$.

Let $\Sigma_p$ be the set of lines in $\mathbb{P}^3$ containing $p$ and let $\Sigma_H$ be the set of lines in $\mathbb{P}^3$ lying in $H$.

Then under the Plücker embedding $\Sigma_p$ and $\Sigma_H$ are carried to two linear subspaces of dimension $2$ in $X$ such that

$\Sigma_p \cap \Sigma_H= \emptyset$.