For problem 1, since is finite, we have , so the condition can be rephrased as . I would try to prove by contradiction: suppose there is a another subgroup such that . Then I would consider the quotient group and try mapping into it, using the canonical homomorphism.
For problem 2, we can assume that and are finite. Let the collection of left cosets of in be and the left cosets of in be . It would suffice to show that the set is the collection of left cosets of in .
Can you give that a shot?