Question 1. Let $I(Y) \subset \Sigma(Y) \times \Sigma(Y) \cong \mathbb{P}^2 \times \mathbb{P}^2$ be the incidence scheme (parameterizing pairs of intersecting lines). Then $I(Y) \cong \mathrm{Fl}(1,2;3) \subset \mathbb{P}^2 \times \mathbb{P}^2$; I think you can find this in Sanna, Giangiacomo. Small charge instantons and jumping lines on the quintic del Pezzo threefold. Int. Math. Res. Not. IMRN 2017, no. 21, 6523-6583. In particular, lines intersecting a given line $L$ are parameterized by $p_1(p_2^{-1}([L])) \subset \Sigma(Y)$ which is indeed a line on $\mathbb{P}^2$ (here $p_i$ denote the projections of $I(Y)$ to the factors).
Question 2. Recall that $Y \subset \mathbb{P}^6 = \mathbb{P}(V)$. In particular, every line on $Y$ is a line in $\mathbb{P}(V)$. This defines an embedding
$$
\Sigma(Y) \to \mathrm{Gr}(2,V).
$$
It is defined by a rank-2 vector bundle $\mathcal{U}$ on $\Sigma(Y)$. A description of this bundle can be found in the same reference, for now it is important that $\det(\mathcal{U}) \cong \mathcal{O}(-3)$. A quadric in $Y$ is cut out by a quadric in $\mathbb{P}(V)$; its equation is in $S^2V^\vee$, and it induces a global section of $S^2\mathcal{U}^\vee$. The tangency locus is the degeneracy locus of the corresponding section of $S^2\mathcal{U}^\vee$, or equivalently of the induced morphism
$$
q \colon \mathcal{U} \to \mathcal{U}^\vee.
$$
Its equation is $\det(q) \colon \mathcal{O}(-3) \cong \det(\mathcal{U}) \to \det(\mathcal{U}^\vee) \cong \mathcal{O}(3)$; thus the the tangency locus is a sextic curve in $\Sigma(Y) \cong \mathbb{P}^2$. For general $q$ it is smooth, but it is not true that it is smooth for any smooth quadric divisor --- if, for instance, a divisor contains a line, this line is contained in the tangency locus and gives a singular point on it.
