Your use of not using "to" before "home" as it is an adverb is correct, as well as your second statement. That use of "You come to home" is really, really strange and I've never heard that before. It seems to be a typo. I would say "You come home and you want to tell your brother what your friend told you." Either that or "You come to your home...", though the first one is much more natural.
Anyways, to answer your question, here are a couple of SE posts!
https://english.stackexchange.com/questions/33652/why-is-to-not-used-before-home
This one details how the word "home" functions in several ways (how it can be either a noun or an adverb).
Can we use "to" before home, if we are using determiners (her, my, your etc.) before home?
The second explanation is very, very good. It details how "home" acts when bare and surrounded by other things.
I hope this helps, and feel free to ask if you have any more questions or if I didn't answer properly! :)
I'm a native English speaker, from the Midwest (US), if that counts for anything. There are variances in language, so if this is a regional thing, I don't know about it and I apologize for that.
Prepositions can be tricky because they often have multiple meanings depending on context.
"By" can mean "near", as in, "The house is by the water." It can also mean "using", as in, "We travelled by car." That works here. "We shared by email" means "we shared using email".
"Through" usually means "passing over the boundaries of", as in, "We drove through Spain". It carries the implication of going in and coming back out -- you don't use "through" when you stop inside the place or thing. You might use "into".
We sometimes use "through" to describe a method of communication, I guess on the thinking that the message travels "through" this communication system. Like, "I sent a letter through the mail", as in, the message travelled through the postal system. Or, "We kept in contact through the telephone." So in that sense you can say, "We shared through email", i.e. the sharing went through the email system.
Note this usage is pretty much limited to communication. You wouldn't say, "We travelled through train", you'd say "We travelled by train." ("We travelled through a train" would mean that you walked up and down the length of the train, or that you walked in one door of a train and then out the opposite door, not that the train carried you somewhere.)
Best Answer
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.