"No, I am not Arab."
Translation:Nem, én nem vagyok arab.
I'm pretty sure "nem arab vagyok" should also work, but I'm wondering what the difference would connote. Is there a difference in emphasis?
Yes. You would phrase it like that if you were then going to contrast it, something like:
"Nem arab vagyok, hanem görög." - "I'm not Arab, but Greek."
Can the "én" also be dropped from the given translation? As in "Nem, nem vagyok arab"? Or does the word order also need to be changed to put nem before arab.
Your sentence is correct. The én can be dropped and nothing needs to change. The nem usually stays in front of the verb unless you want to put the emphasis on the fact that you're not Arabian, but something else. Iraqi or so.
An example of when arab comes before nem:
- Nem, arab nem vagyok, viszont az egyik dédnagyapám kínai volt. = "No, I am not Arab, but one of my great-grandfathers was Chinese."
Is "vagyok" required here because it is a negative sentence? (And if it were an affirmative sentence, would it be correct to just say "én arab?")
Whether it's a positive or a negative sentence has no influence on whether you use vagyok or not. The trick is this: when you have a sentence saying "I am / you are [something]", then you'll have to express it with a form of van, except for the third persons.
Let's try it with my all-Arab family:
Arab vagyok. Te is arab vagy. Ő arab. Arabok vagyunk. Ti arabok vagytok, és ők is arabok.
Generally I would van rather translate with "there is" and vannak with "there are", since those words are not used for quality descriptions ("The car is red"), but for matters of placement and existence ("There is a car behind the house.")