"Do you have a phone?"
Translation:너는 전화가 있느냐?
Yes. Unless you have been talking about another person before and need to indicate the change of topic, or if you’re making a contrast between 너 and somebody else. For example: 아, 난 스마트폰을 집에 놓고 온 것 같아… 너는 전화 있느냐? “Oh, I seem to have left my smartphone at home… Do you have a phone?” (Contrasting “you” vs “I”).
It's an interrogative verb ending, i.e. one which forms a question. It's also quite blunt (I'd say even more so than if you use the -아/어 form), so only use it on people you don't have to show any respect to, never superiors and not on equals either unless you're really close friends – or trying to pick a fight.
Yes, although in my experience it is more common in historical settings. In modern ones you are more likely to see -냐. Still, I would advise against using it except to children and maybe the very closest of friends as it tends to sound like you’re talking down to somebody.
전화 is used for both the physical phone itself and for “phone call”. This is one difference to 핸드폰 which only refers to the device (i.e. you can't say 핸드폰하다). The other difference is that 전화 is a more general term which can refer to cell phones as well as mobiles. 핸드폰 on the other hand only refers to cell phones.