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Yes, it does.
It has rotational symmetry of order 2.
No, it does not. Take a sheet of A4 paper and fold it across the diagonal, and you will see that the two halfs do not line up.
The other name for the line of symmetry of a circle is the diameter. The diameter of a circle is a line segment that passes through the center of the circle and has endpoints on the circle itself. It divides the circle into two equal halves, making it a line of symmetry. This means that if you were to fold the circle along the diameter, both halves would perfectly overlap.
Yes. An ellipse (oval) has two lines of symmetry, but not a rotational symmetry. A parabola has one line and no rotation.