Smallest possible habitable planet? (also taking density into account)

I have seen a few similar questions but none seem to take the density of a planet into account. I'm creating a planet on which several small civilisations develop. The planet needs to have a few biomes of varying temperature such as mountain ranges, deserts, temperate areas etc.

What is the smallest this planet can be? Also taking into account that this planet may be many times the density of Earth. For instance a planet half the volume but of double the mass would have the same gravitational pull (correct me here if I'm wrong). Would this higher density cause any problems as well?

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Smallest possible habitable planet? (also taking density into account)

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