I think this is because in English, "coffee" is not counted. That is, you cannot say "a coffee" in English because it is not a definite quantity. You can say "a cup of coffee" instead. This is one of the instances where translation between Arabic and English do not exactly match (specially in the definition of some words).
Another example, you might say in English: I like nature (without defining nature because it is a general term and you are talking in a general sense). But in Arabic, it doesn't work that way. You would say it as أنا أحب الطبيعة, where الطبيعة is (the nature); It comes defined with "AL" and it would make no sense for an Arabic listener to hear it in this sentence without AL.
Absolutely right, that "coffee" is a non-count noun. But in oral speech it can be a count noun; people say "a coffee" to mean "a cup of coffee". In addition, you don't need to go as far away as Arabic to find a definite article with, eg, "nature". In French, it's exactly the same: "j'aime la nature". In Russian, where articles don't exist, it's much simpler, no choice! At least in Arabic, there's only one gender, as far as I know, for the article. In French of course there are two. And if Russian had them, there would be three! Perhaps that's why they abolished them. Joke.
English is not my first language but it seems to me that "Arab" is used for the nation or an ethnic group, in the same manner as "Turk" is used. Likewise, it is typical to say "Turkish coffee" and not "Turk coffee" - so I guess for the Arabic-part it should be Arabian or Arabic coffee, but not Arab coffee, I presume.
Yep - this is the correct way to say it.
if the sentence goes on and some words come after, then we complete the last vowel in (3arabiya) and it becomes (3arabiyatun) - to be precise, it should be (3arabiyya) and (3arabiyyatun). There is Shaddah on ي (Y), so it comes as a double.