Its simply called the "center point" or "center". There is no fancy term for it.
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The diameter of a circle passes through the center of a circle at its center point
diameter
No. Every circle on the sphere whose center is also the center of the sphere is a great circle. If the circle's center is not also the center of the sphere, then the circle is a small circle.
The line from the center of a circle to the edge is called a radius. The radius can be used in pi and a lot of other things. If you use the radius the wrong way, your answer will be wrong. Most people call it the " line from the center of a circle to the edge of a circle " but it's called the radius.
The line from the center of a circle to a point on the circle is the radius.