First question: both are correct Second question: A contextual sentence would help. I was in the bath at the moment it happened.
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Mister Micawber.
These are all OK:
In/At the moment in/at which it happened, he immediately ran away-- as Philip indicated, 'at' is much better for a moment-- which is only momentary. This one also seems vaguely redundant.
In/At the moment when it happened, he immediately ran away
In/At the moment it