For question 1, does your multiplication in the group simply mean operation in ? If it is, you can consider the closed interval in , with . Then the associative law holds and inverse exists in , but is not a group as .

For question 2, my first thought is to consider a regular -gon with operations reflection about an axis and rotation by angle . The reflection and rotation forms a group. The reflection is a subgroup of order 2. Check that reflection before rotation does not give the same if we reverse the order of operation. I am not sure if we can always have this statement. Please check for me.