For (b), you need to prove that these elements form a subgroup (as you will get normality for free because they are closed under conjugation). To prove that it is isomorphic to the Klien 4-group, you just need to show that it is not cyclic (there are only 2 groups of order 4, and if it is not one it must then be the other).
For (c) you need to calculate the set . This is a group because :
Now, as you have calculated the set you know it has order 12, and that it contains at least two conjugacy classes (the identity, and the one in part (a)). The third will just be the rest of the group. You should prove this is a conjugacy class.
Now, neatly, you have that your group is also normal as it is a union of conjugacy classes so it MUST be normal.