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What drove the expansion 250 million years ago is pure speculation but it suggests there is a history for such events.
How would you explain the stability of the Moon's orbit under a regime of an Earth substantially increasing in mass? How do you explain away the palaeomagnetic data? Mountain ranges? Subduction zones and so on?
It is not easy to estimate how far away from the Earth the Moon was when it formed, but simulations suggest is was about 3-5 times the radius of the Earth, or about 19-30 thousand km. (The Moon is currently about 384,000 km away from Earth or 3-4 thousand times further away than this.) The Moon probably couldn't have formed closer than 3 Earth radii because tidal forces from the Earth would just pull it apart again, and it is unlikely that the impact could have ejected material further than 5 Earth radii. It's not a totally easy questions to answer though as it depends a lot on the (unknown) details of the impact and how the hot material behaved in space.
The exact rate of the Moon's movement away from Earth has varied a lot over time. It depends both on the distance between the Earth and the Moon, and the exact shape of the Earth. The details of continents and oceans moving around on Earth actually change the rate, which make it a very hard thing to estimate. The rate is currently slowing down slightly, and it is estimated that in about 15 billion years the Moon's orbit will stop increasing in size.