Not the same, you're right. But not quite like an "ö" either. Here it is also pronounced differently: http://forvo.com/word/vad_g%C3%B6r_du_i_sverige%3F/#sv
Yes, g has two pronunciations, one 'hard' as g (like in 'guard') and one soft as you might think of as y, like in 'yes'. The soft pronunciation occurs before the vowels e, i, y, ä, ö. There are several useful links about pronunciation in this thread: https://www.duolingo.com/comment/5892805
We never use är together with another verb in the present, and neither do you in English. Saying du är gör is like saying you are do.
In questions, the verb must go before the subject, and the only thing that can go before the verb is a question word.
This leaves you with Vad gör du? as the only possible option.
Think of the question "What have we here?" Swedish "What do we have" would be like "What have we?" "Vad har vi?"
After a question word, we need a verb (just like in English, except we use so and are/am/is as auxiliary verbs). Translate that sentence into English: What you have? It sounds just as wrong in Swedish. Now swap the verb and pronoun: What have you? It sounds old fashioned but much better, since after a question word we expect to hear a verb.
In principle it should, but it might be less necessary in Swedish than in some other languages because it's normally clear anyway from word order whether it's a question or not. If you do ask a question by using a sentence that doesn't have question word order, you have to make it clear by intonation, so in those cases it's really important.