it's hard to say what the order of g^{12}might be, as we have no way of knowing just from |G| = 100, what |g| even is. the best we can do is just that g^{12}is some divisor of 100. if G is cyclic, we can say more, because cyclic groups are more "predictable". if G is merely just abelian that is somewhat helpful, because abelian groups are direct products of cyclic groups.

as to your other question, clearly A∩B has to have an order which is a common divisor of |A| and |B|, which leaves us with 1,3,5, or 15.