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If we solved GR in the case of two black holes colliding, does each individual black hole have a single pointlike singularity at their respective centres, and would those singularity combine?

Trying to visualise this in my head, I see two black holes moving towards each other, combining and then releasing energy as gravitational waves.

Two things that strike me with this model is that

1) This doesn't look very time symmetric. (So do we need to bring in theories of thermodynamics and entropy to explain the time asymetry?)

2) If each black hole had a singularity at its center then there would be a discontinuity if the singularities combined.

So, with pure GR, would black holes actually combine at all or just either bounce off elastically, or pass through each other?

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