The unit circle is a circle that can be used to find trigonometric functions.
The equation of the unit circle is x^2 + y^2 = 1.
So it is any circle with radius 1.
No, because if a circle has a set amount of sides, then it must have a certain amount of angles, reguiring the vertex of the angle to be farther from the center than say,the midpoint of its adjacent...
In a sense it's entirely arbitrary, but 360 has the advantage that it can be divided by so many different numbers (technically it has a lot of factors).