I am writing up my comments as an answer. Let $f:\widetilde{X}\to X$ be a minimal desingularization of $X=X_n$. For a general member $\mathcal{C}$ of the family of curves $\mathcal{C}_{p,q}$, let $\widetilde{\mathcal{C}}$ be the strict transform of $\mathcal{C}$ in $\widetilde{X}$. Then $\widetilde{\mathcal{C}}$ is a smooth, projective curve of genus $0$ on a smooth surface. Moreover, since we can vary $p$ and $q$ to any sufficiently general pair of points, the curve $\widetilde{\mathcal{C}}$ moves in a $2$-parameter family of deformations on $\widetilde{X}$. By the classification of surfaces (in characteristic $0$, of course), this forces $\widetilde{X}$ to be a rational surface, and $\widetilde{\mathcal{C}}$ is a nef and big divisor. In fact, Proposition 2.3.3 of Shen's thesis implies that the complete linear system of $\widetilde{C}$ is basepoint free, the associated morphism to projective spaces restricts on some Zariski open neighborhood of $\widetilde{C}$ to be an embedding, and the image of the contraction is a normal surface of "minimal degree". Thus, by the del Pezzo-Bertini classification, the image $X'$ is either a smooth rational surface scroll, a Veronese $2$-uple surface, or a cone over a rational normal curve.
Anyway, consider the restriction of $f^*\mathcal{O}(1)(-r\widetilde{\mathcal{C}})$ to $\widetilde{\mathcal{C}}$. This is an invertible sheaf on a smooth genus $0$ curve that has degree $n-rm$, where $m$ is the self-intersection number of $\widetilde{\mathcal{C}}$ on $\widetilde{X}$. Notice, $m$ is positive since the linear system is ample. Thus, the total degree is negative for $r>n/m$. Since $\widetilde{\mathcal{C}}$ gives a basepoint free linear system, it has nonnegative intersection number with every effective divisor. Thus, $f^*\mathcal{O}(1)(-r\widetilde{\mathcal{C}})$ has only the zero section when $r>n/m$. That means that we can bound the dimension of the vector space of global sections of $f^*\mathcal{O}(1)$ by the sum over $r=0,\dots,\lfloor n/m \rfloor$ of the dimension of global sections of $f^*\mathcal{O}(1)(-r\widetilde{\mathcal{C}})|_{\widetilde{\mathcal{C}}}$.
For an invertible sheaf of degree $d$ on $\mathbb{P}^1$, the dimension $h^0$ equals $d+1$. Thus, the sum above evaluates to, $$N+1 \leq (\lfloor n/m \rfloor + 1)(n+1 - (m/2) \lfloor n/m \rfloor).$$ This is maximized when $m$ equals $1$. This gives the bound, $$N+1\leq (n+1)(n+2)/2, \ \text{ i.e., } N\leq n(n+3)/2.$$ Thus, Question 2 has a positive answer.
It is possible to get a classification as in Question 1, but it might not be very explicit. By the computation above, $m\leq n$. Thus, by Mingmin Shen's theorem, the surface $X'$ with hyperplane section $\widetilde{\mathcal{C}}$ is a surface of minimal degree equal to $m$. For each of the finitely many such minimal degree surfaces of degrees $m=1,\dots,n$ in the del Pezzo-Bertini classification, we can list all of the linear systems whose intersection number with $\widetilde{\mathcal{C}}$ has degree $n$. Then we consider sublinear systems of that complete linear system with specified basepoints (basepoints of the linear system on $X'$ give negative self-intersection curves on $X$).