On a lower bound for the birthday problem
The question's $\displaystyle C(q,N) \ge \frac{(q-1)^2}{2N}$ bound is wrong. That's incompatible with the well-known fact that for $N=q^2$ and large, $C(q,N)\approx1-e^{-1/2}\approx39.3\%$.

A correct bound is $\displaystyle C(q,N) \le \frac{(q-1)\,q}{2N}$, valid for all $q$ and $N\ge1$, and tight when $q\ll\sqrt N$.

Another correct bound is $\displaystyle C(q,N) \ge \frac{(q-1)^2}{4N}$ when $1\le q\le\sqrt{2N}$, which follows from the question's lemma.

See this answer for some derivations, but beware that the notation there is $(n,k)$ and $p_n$ where the question has $(q,N)$ and $C(q,N)$.