This is related to Ueno's Algebraic Geometry 2, Chpt 6, Thm 6.25.
Let $X$ be a non-singular algebraic surface. $C_1,C_2$ 2 algebraic curves over $X$ do not have a common component. Then $C_1\cdot C_2$ intersection is defined by $\chi(X,O_X)+\chi(X,O_X(-C_1-C_2))-\chi(X,O_X(-C_1))-\chi(X,O_X(-C_2))$.
It is not too hard to see the following exact sequence is exact.
$0\to O_X(-C_1-C_2)\to O_X(-C_1)\oplus O_X(-C_2)\to O_X\to O_{C_1\cap C_2}\to 0$
$O(-D)$ is the ideal sheaf associated to curve $D$. The first map on affine chart is obviously given by $x\to (x,x)$. The second map is given by difference for comparison.
$\textbf{Q1:}$ The book says above exact sequence follows from $0\to O_X(-C_1)\to O_X\to O_{C_1}\to 0$ (1) and $0\to O_X(-C_2)\to O_X\to O_{C_2}\to 0$ (2). My guess is considering $(1)\otimes_{O_X}O_{C_2}$. Then I have $O_X(-C_1)\otimes O_{C_2}\to O_{C_2}\to O_{C_1\cap C_2}\to 0$. Now $O_X\to O_{C_2}$ is surjection. The only trouble is to find out kernel of total composition $O_X\to O_{C_1\cap C_2}\to 0$. I do not see how to splice sequence to obtain the exact sequence. However, given the sequence above, I can easily check exactness.
$\textbf{Q2:}$ When is $X$ non-singularness used in the computation? Computation of $\chi(X,O_X)$ uses only Cech covering. This computation does not seem to invoke non-singularness of $X$. Ueno has done cohomology computation of $P^n_k$ over a field $k$ so far. However, it does not seem to be the case that the computation truly invokes $P^n_k$ smooth, though smoothness is already tangled with polynomial ring structural locally.