I assume that your definition of "subgroup" is "a subset which is a group under the operation of $G$."
For problem 1, you just need to show that the conditions imply that $H$ satisfies the axioms of being a group; you will need part 2 to show that multiplication restricts to an operation on $H$; then use part 1 to get that $H$ has an identity; and part 3 gives you the inverses. The converse (that if it is a subgroup then it satisfies these conditions) should be easy.
For problem 2, showing that if $H$ is a subgroup then these two conditions hold should be easy. To show that the conditions imply $H$ is a subgroup, try to show that it satisfies the three conditions in problem 1. Since $H$ is nonempty, there is an element $x\in H$. Now apply the condition with $g=h=x$ (note that we do not require $g$ and $h$ to be distinct!) to conclude that $H$ contains the identity. Then take $h\in H$, and set $g=e$ to conclude that if $h\in H$ then $h^{-1}\in H$. And finally, given $g,h\in H$, use $g$ and $h^{-1}$ to show that $H$ is closed under products.