Learn English – You slept well AT night or IN night

Well, first off it isn't an adverb. I'm not sure if you tagged the question yourself or if somebody did it for you, but it is indeed a preposition.

Otherwise, your example sentence is not incorrect, but it's also not the most natural phrasing. Most speakers would be more likely to say "What about the book under that one?" The prepositions "under"/"underneath," and "below" are fairly similar in their dictionary meanings, but are generally used in different ways.

Under is used when something is directly underneath another object, especially if it's being covered or hidden. A man would conceal a weapon under his coat, during a storm you may hide under a table, or perhaps you could drop your keys and lose them under the seat.

Below, on the other hand, simply indicates relative position and is more likely to be used when the two things in question are not immediately next to each other. Submarines cruise around below the surface of the water, my friend lives in an apartment a few floors below mine, and the molten core of our planet is far below the surface.

Although those are examples in which below is preferred, you can usually use under instead without sounding at all unnatural. The reverse is not always true. So, when in doubt, stick with under. Learning to use below in a sentence which sounds natural to native speakers is a skill that will come with time.

A brief note: Below is also used when imagining a sort of.. non-physical "vertical scale" rather than talking about actual positions. I could say a family lives below the poverty line, that the temperatures tonight may drop below freezing, or that a person is of below average intelligence. In this context it would be very bizarre to use under instead.

First off, prepositions are tricky words, and it's hard to talk about all the ways they can be used.

That said...

at

We can use at night to mean during the nighttime:

I was there at night.

(notice how I did not say "at the night," which is what you wrote in your question).

in

We can use in the night to mean during the nighttime or at various times during the night:

Hyenas prowl in the night. (at night would work, too)

into

We use into the night to say that something continues for a long time after sunset:

The partygoers reveled late into the night.

on

You are correct, we don't usually say on night or on the night. However, if we expand the phrase, and stipulate which nights we are talking about, then "on (the) nights when..." is a relatively common construct:

The band liked crashing at our place on nights we played at Live Bay.
I try to go to bed early on Sunday nights.
On the nights when we hear the ice cream truck, we'll go outside and buy ice cream.

Now, let's combine some of these together:

We usually lock the front door at night. However, our roommate Micheal is an actor, and he sometimes works late into the night. On the nights Micheal hasn't come home before midnight, we leave the front door unlocked, in case he has forgotten his key.