The locus of lines intersecting with another fixed line on a Fano threefold

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 $S^2mathcal{U}^vee$. The tangency locus is the degeneracy locus of the corresponding section of $S^2mathcal{U}^vee$, 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.