I have many questions starting with why for hungarian :-P

Because of… history, maybe? Maybe it's just a coincidence. It also happens with vár (meaning both "to wait" and "castle")…

"vagy" first meaning is "or" second meaning is "you are" in singular

They should be exactly the same. And they sound exactly the same to the Hungarian ear. That is, the difference you are hearing is not significant in the Hungarian language. Yes, it is very soft in this audio but it has no significance. It is just how it came out of the speaker's mouth in this case.

The short "a" in Hungarian is fairly far back in the mouth, so it can sound a bit like an "o" in other languages. Also, the "gy" is palatal, so if you're not fully hearing the stop and release of the consonant, you still might be hearing the movement of the tongue to a palatal position, which is where /j/ is pronounced ... so hearing it as "voi" is actually pretty understandable.

"gy" is always pronounced the same way. Here it may sound 'softer' because it is followed by a consonant

In this case 'vagy' means 'or', and it is not used as a form of the english 'is'.

Think of 'Igen vagy nem' as a way of somebody pushing for an answer. "Do you want this ? Come on give me a straight answer. Yes or no?"

The word "not" is a negative adverb. "Yes" is an interjection, as are "Wow!" or "Ouch! " So the word "no" in the question "No, or yes?" would be an interjection of negation, like "Ewwwww!" (in "my" English). Consider: "Hooray, or boooooo?" - "Igen, vagy nem? [And no! "nem" is not a negative quantifier in this context. (Or could it possibly be so by implication - "Magyarul"?]

I have a "grammar-contrary", but creative, friend who, regularly, does (or, at least, he tries to) perempt and pre-empt my proposed or continuing course of action with that obnoxious expression: "Not!" But no! I am not the oppositional one. Someone please correct me if I am wrong!