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.
The issue here is that "way" has a large number of meanings.
In the first sentence (which by the way is the only one where I could use "in the way" though it tugs at my ears) you have
4
a : manner or method of doing or happening < admired her way of thinking>;
also : method of accomplishing : means < that's the way to do it>
The second sentence uses the same meaning but in a different way. Changing it around to
The way she manages to stay calm is amazing.
Shows you exactly why you couldn't have "in the way" there. Really it just takes the meaning "how" I can't find that exact meaning in the Merriam-Webster but I would say it really boils down to "method of accomplishing".
The third sentence uses "the way" idiomatically the M-W has the following explanation and example
The last version uses a different meaning again
9 : state of affairs : condition, state
The only case of these where you can use "in the way" is when you are using way as a synonym for "manner" or "method".
"in the way" has many more possible meanings derived from the meanings of "way".
As examples consider
We couldn't move further; the blockade was in the way.
Meaning it was obstructing further progress.
She's is in the family way.
Meaning she is pregnant.
Best Answer
Of and about are by far the most common, and mean essentially the same thing.
In fact, when we look specifically for "I'm thinking * buying", any other choice is so rare it might as well not exist. The author you quote as writing "I'm thinking on buying..." might be using a regionalism.