one plane LINE
In plane geometry there is exactly one straight line through two points. There can be any number of curved lines.
between two point there is exactly one line between three points there is exactly one plane
I think you mean: Are any three points contained in exactly one plane? only if they're not collinear... I think
Through any three noncollinear points, there is exactly one plane.
No, it is not true. Just think of the three vertices of a triangle.