1. Don't call me names.

2. He is calling me a name.
3. She is calling him names.
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Are they all grammatical?
Which ones are commonly used?
Does 'call somebody names' mean 'speak ill of others'? Why is that?

We can see good expressions as follows?
He calls me Bob.
He calls me Strong.
He calls me Prince.

You are right. The phrasing "calling him names" and others that are similar means being derogatory or saying negative things about others. Since that's a colloquial term, I don't know how it got started.

Your examples of good expressions are correct, too, except I would not use capital letters for "strong" or "prince."

1. "Don't call me names" is a grammatically correct sentence. It is a command or request asking someone not to use derogatory or offensive terms when referring to the speaker.

2. "He is calling me a name" is also grammatically correct. It describes an ongoing action where someone is currently using a derogatory or offensive term to refer to the speaker.

3. "She is calling him names" is also grammatically correct. It describes an ongoing action where someone is using derogatory or offensive terms to refer to a male person.

All of the sentences are commonly used in English language communication, especially the first two. The third sentence may be less common, but it is still used to describe situations where someone is insulting or belittling another person by using derogatory terms.

The phrase "call somebody names" indeed means to speak ill of others using derogatory or offensive terms. It can be used to refer to any situation where someone insults or belittles another person by using inappropriate language or derogatory terms to describe them. This phrase emphasizes the act of using words to hurt or demean someone verbally.

As for your examples, they are all correct expressions using the phrase "He calls me." They demonstrate different ways someone might refer to you. For example, "He calls me Bob" means that he uses the name "Bob" to refer to you. Similarly, "He calls me Strong" and "He calls me Prince" mean he uses those terms to address you, perhaps as a nickname or a way of characterizing you.