1. None
My book and dictionary say that I can use sigular and plural verbs(both) for 'none'. But it seems like native speakers use plural verbs more. Is there any reason? The example sentence was; None of his sons live(s) with him. And they said 'live' is correct in the sentence.
2. Neither
When you see the sentence; Neither man trusts the other, I thought 'Neither men trust each other.' is right. Is this sentence right? How about 'Neither men trust the other.'?
Here's the simple difference between 'none' and 'neither' for beginners.
None = not any
Neither = not any of two options
Examples
A: Do you want to go at 9am, 10am or 11am? (More than two options)
B: None of them.
A: Do you want to go at 9am or 10am? (Only two options)
B: Neither of them.
Let's look at your questions:
None
The sentence is correct: "None of his sons live with him."
The reason is that 'none of his sons' are considered a singular group (like the pronoun 'they'), rather than individual people. Therefore, the correct verb is 'live'.
Other examples of a group being used in the singular are:
None of the drinks are alcohol-free.
None of them are alcohol-free.
They aren't alcohol-free.
None of the cats eat fish.
None of them eat fish.
They don't eat fish.
The plural form is used when the 'none' is incorporated into the subject of the sentence.
The singular is used when 'none' becomes the object.
There is none left.
We have none.
I ate none.
However, some of these sentences would be a little too casual for me. I would prefer:
There aren't any left.
We don't have any.
I didn't eat any.
Neither
Both of your sentences are correct.
"Neither men trust each other."
The statement reflects the view of both men at the same time: A and B do not trust each other.
"Neither man trusts the other."
Here, the statement reflects the view of each man: A does not trust B, and B does not trust A.
I hope that helps.