I think it puts further emphasis on ahora or now. Basically it's the difference between 'right now' and just 'now'.
Yes, "mismo" usually means "same". Here, "ahora mismo" literally means "now itself".
What is the correct sentence? The suggested "What have you got to do right now?" or "What do you have to do right now?" Is there any difference?
I think they're both technically correct but it depends on where you live (especially in the US), I'm from the northeast and we rarely say "what have you got to do right now", we'd say the 2nd sentence you suggest. I think in the southern us they use "have you got to do" more often. To me "you got" sounds strange even though it's not technically wrong.
Your second option was accepted 13/7/18 - but why cannot it be "What must you do"?
Mismo is the right in the English phrase right now. It has nothing to do with correctly. It is used to emphasize now. Right now = immediately, or at this time, at the moment.