Do black hole singularities actually merge?
Infinities are hard to bend the mind around, but in this case, the merge is not impossible. Yes, the distance between them must reach zero in order to make the black holes merge, but the rate of which energy is lost to gravitational waves also increases when they get close to each other.

We are therefore dealing with a $\frac{\mathrm{infinity}}{\mathrm{infinity}}$ problem, where a finite limit may, and in this case does exist.

Do not forget your $\lim$'s!