This is rather late since the question has long since been settled, but I wanted to make the following comment (which is too long for the comment box): when it comes to questions of this sort, toric varieties never have infinitely many of anything.

Less flippantly, what I mean is the following: for a variety $X$ of any dimension, generalising the question about the number of $(-1)$-curves, one can ask about either

the number of extremal rays of the
cone $Eff(X)$ of effective, or

the number of extremal rays of the cone $Nef(X)$ of nef divisors.

(In the surface case, every $(-1)$-curve spans an extremal ray of $Eff(X)$, and corresponds by duality to a codimension-1 face of $Nef(X)$.)

Now my point is just that for $X$ a toric variety of any dimension, both cones $Nef(X)$ and $Eff(X)$ are known to be closed cones spanned by a finite set of vectors. Indeed, there is the following statement, due to Cox, which lncludes those two statements:

**Theorem (Cox):** The Cox ring of a toric variety is finitely generated.