The Difference between at night and by night is that the former shows time (' We work during the day and sleep at night'), whereas the latter suggests method, or the conditions under which something is done or takes place (' Burglars and poachers work by night', 'In order to avoid the heavy traffic on the roads, some people prefer to travel by night').
The difference between at night and in the night is that at night is used when talking about what generally or regularly happens then (' The noise of the traffic keeps me awake at night'), whereas in the night is used when talking about an event that happens at some time during the night ('There had been heavy rain in the night').
On is used as in on the night of 10 January/January 10, (on) Saturday night, (on) that night, etc.
Night after night expresses the idea of a succession of nights taken individually (' Night after night she sat by the bed of the sick child').; night by night, on the other hand, represents the succession of nights as one continuous period or series (' Night by night the policeman paces the streets'). Cf. day by day.
▶️ (1) During the night: e.g. to travel over night. Hence an over-night journey.
▶️ (2) For the duration of the night.
We stayed over night at a hotel.
Cf. Over the weekend, over Christmas, over the New Year, etc.