Define two unit vectors $v$ and $w$ as follows:
$$
v={C_{2C}-C_{1C}\over|C_{2C}-C_{1C}|},
\quad
w=C_n\times v.
$$
You already know how to compute $a$ and $h$, so intersection points are given by:
$$
C_{1C}+av\pm hw.
$$
I don't know a specific reference for thid sort of question, so let me just address your two example questions. (Everything I say is discussed in the books of Kollár--Mori, Lazarsfeld, Debarre, etc.)
Let $X$ be the blowup of $\mathbf P^2$ in 9 very general points. Then the Mori cone of $X$ is well-known: its extremal rays are
the class $-K$ of the anticanonical divisor (i.e. the proper transform of the unique cubic through the 9 points)
the 9 exceptional divisors $E_1,\ldots,E_9$
the images of the $E_i$ under the action of the Cremona group: it doesn't matter what these look like, only that there are infintely many of them.
We can write any class on $X$ in the form
$$ aH + \sum_{i=1}^9 b_i E_i $$
for some $a, \, b_i \in \mathbf Z$, and intersections with the exceptionals are given by $( aH + \sum_{i=1}^9 b_i E_i) \cdot E_j = -b_j$. (For classes of the third kind above other than the $E_i$ themselves, the coefficients $b_i$ are therefore nonpositive.
Since there are infinitely many classes of the third kind above, the corresponding coefficients $(b_1,\ldots,b_9)$ cannot be bounded, and so the answer to
Are the distinct intersections bound, for example to be less than or equal to 1?
is no.
On the other hand if we blow up 8 or fewer points in $\mathbf P^2$, we get a del Pezzo surface whose Mori cone has finitely many extremal rays. That means that of the infinite set of classes above, there are only finitely many in which one of the coefficients $b_i$ equals zero. So back on $X$ a given exceptional $E_i$ has nonzero intersection with all but finitely many extremal rays. Therefore the answer to
Can an extremal ray have non-zero intersection with arbitrarily many other extremal rays?
is yes.
Finally let me mention an obvious example in which one has positive answers to your questions: if $X$ is a nonsingular toric surface, then any extremal ray of the Mori cone is spanned by a torus-invariant curve. Two such curves $C_1$ and $C_2$ intersect as follows:
- if the corresponding rays span a cone of the fan of $X$, then $C_1 \cdot C_2=1$;
- otherwise $C_1 \cdot C_2=0$.
So intersections are bounded by 1, and each ray has nonzero intersection with at most 2 other rays.
Best Answer
If I understand your problem correctly, this is exactly the same as Turán's brick factory problem. There is a conjecture Zarankiewicz's Conjecture which states the number to be $$\left \lfloor{\frac{n}{2}}\right \rfloor \left \lfloor{\frac{n-1}{2}}\right \rfloor \left \lfloor{\frac{m}{2}}\right \rfloor \left \lfloor{\frac{m-1}{2}}\right \rfloor $$ for a complete bipartite graph $K_{m,n}$.
For the complete graph there is a related conjecture stating the number to be
$$\frac{1}{4}\left \lfloor{\frac{n}{2}}\right \rfloor \left \lfloor{\frac{n-1}{2}}\right \rfloor \left \lfloor{\frac{n-2}{2}}\right \rfloor \left \lfloor{\frac{n-3}{2}}\right \rfloor .$$