What mechanism(s) prevented the gravitational effect of the earth on the moon from shattering the moon when it was closer to the earth than the Roche Limit some 4.5 billion years ago?
The leading theory for formation of the Moon is that a large object collided with the Earth and threw off a cloud of debris. This cloud then clumped together under it's own gravity to form the Moon. Simulations of the cloud show it formed outside the Roche limit - at around 1.3 times the Roche limit in fact.
So the Moon has never been closer to the Earth than the Roche limit.