Arrogance can be perceived as an offensive display of superiority. We can have straight forward manifestations of arrogance but also comes in a disguised or cloaked form.
2. How do you detect arrogance?
If it's straight forward it is easy. But sometimes is well concealed behind actions, irony or even gentle words. I believe that we detect (or display for the matter) arrogance from a set of behavioral traits more than anything else.
3. Why some (we?) people become arrogant?
Because we are not wise.-
The Socratic paradox goes like "I know one thing: that I know nothing". Socrates believed that he is the wisest man in Athens because he knows that he does not know while others falsely believe that they know. Pythia (the famous oracle of Ancient Greece) confirms that Socrates is the wisest man in Athens.
What Socrates was trying to say gets a lot clearer by Aristotle a couple of decades later, when he writes "It is the mark of an educated mind to be able to entertain a thought without accepting it.". Shakespeare in 1603 A.D. in As you like it writes one of my favorite rhymes: "The fool doth think he is wise, but the wise man knows himself to be a fool."
Combining all the above, we can say that smart doesn't make wise. Wisdom comes through experience and careful unselfish study. Smart is good, shiny but can get us easily out of our way. Wisdom is what we should be aim at.
Then the question becomes: What is wisdom? And Socrates replies "The act of making wise decisions". Which decisions are wise? "The ones that do not hurt your spirit". Socrates believes - like Christ- that even if actively attacked, we should avoid attacking-back because it will ultimately cause us anger and hurt our pure spirit. Socrates says that every human has the obligation to keep his spirit as pure as possible: no anger, no fights, no fears, etc.
4. Is it always bad?
Yes. It is always bad.
5. If so, how do you avoid being arrogant?
See answer No 3.