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When I learned English, my teachers told me that all questions must have an auxiliary verb at the beginning, just like Are you mad? or Is she playing? do.
But when watching some movies or talking with people who speak English, they just ask using things like You mad? and She’s playing?
Of course, the kind of intonation let me know they are asking a question, but why this happen?
They do start with an auxiliary verb, but since it's predictable, it's often omitted.
These are examples of what's called Conversational Deletion in the literature.
The link has references and further examples.
I can think of several valid examples of questions that neither start with an auxiliary verb, nor have been pruned through conversational deletion:
Come again? [Idiomatic question construction meaning "Please repeat whatever it was you just said", or sometimes merely expressing disbelief]
In what way? [Seeking some kind of clarification]
By when? [Seeking clarification of a deadline or other time constraint]
Really? [Seeking confirmation of a purported fact]
Meaning what? [Seeking clarification]
Because real English speakers speak real English, and didn't learn it from your teachers.
Forms like You hungry? and She coming? are common in speech, but are very informal, and would not be found in most written contexts.
Your interesting question has two completely different examples.
The first is a case of Conversational Deletion as described in John Lawler's answer here. This is when we miss out pronouns, auxiliaries and other grammatical words like that in informal speech and writing. It's sometimes called Diary Drop because it often happens in diaries. Many people have studied it in the diaries of famous authors like Virginia Woolf and Winston Churchill.
The second example though is NOT a case of conversational deletion. It is what is sometimes referred to as an Intonation Question. This is when the grammar does not show that the sentence is a question. The structure is the same as it would be for a declarative sentence. We therefore use intonation to show that we are making a question. Again, we normally only do this in informal speech, and not very often. Learners of English overuse this if their first language only uses intonation and not grammar to mark interrogatives.
It is easy to show that there is no conversational deletion here:
COME is an intransitive verb so there is no direct object here. All the parts of the sentence are present and correct. Nothing has been deleted. You can also see that the form of the question is exactly the same as the affirmative sentence: She is coming.
