Exercise
Suppose G is a cyclic group generated by the element g. If g2 also generates G then
G has odd order
G has odd or infinite order, both possibilities can occur.
G has even order
Yes.
No, if G is infinite and g2 generates G
as well, then there exists an m with g = g2m. But this implies that g and hence G has finite order.
No, in that case the gcd of 2 and the order of G is not 1.