The whole thing is complicated and murky. It is very possible that Cruz is not eligible to hold any office in this country.
Cruz is an American. A real one. He got his passport like everyone else.
Contrasted with Barack Obama, who had his first USA passport “conferred” upon him by the US Senate somehow and then eventually had all the records destroyed (oh, and someone died in that little event).
As for Obama his past is so murky who knows what the truth is. I saw a very good story that Obama may in fact be the illegitimate son of Frank Marshall Davis and his mother someone unknown and rest with Obama and Ann Durham contrived. Nothing that can't be cleared up with a DNA test.