a) if an element has order 6, in a group of order 6, it generates the whole group, i.e. this is

b) This would be the lone nonabelian group of order 6. or they are the same.

c) I don't think there is such a group, the sylow 3 subgroup is normal and they have trivial intersection so HK is a subgroup of order 6, so G must be a semi direct product, ie one of the two i mentioned before. Alternatively, cauchy's theorem guarantees the existence of an element of order 3.