Rand's and Galad's mother is Tigraine Mantear who was the daughter-heir of Andor and married to Galad's father Taringail Damodred.
When she disappeared into the Waste after birthing Galad, Taringail then married Morgase Trakand (Elayne's mother) who later became the Andoran queen. Morgase and Taringail had Elayne and Gawyn while they were married.
Meanwhile in the Aiel waste, Tigraine became pregnant with Rand via clan chief, Janduin of the Taardad Aiel.
So there is no direct blood connection between Elayne and Rand as their parents were not in any way related by blood (probably there are a number of distant links over the generations between house Trakand and Mantear but there's nothing concrete in Jordan's writing that I have found, but royal houses usually have numerous connections to other royal houses over a number of generations).
This means that Rand and Elayne's relationship would be legal in the USA and probably anywhere.
As Taringail is not connected to Rand by blood this means that Moiraine (Taringail's half sister) is not related to him. Meanwhile Luc (Tigraine's brother) is Rand's uncle.