For the first one: You can express the expected cost in terms of the parameter (by Law of total expectation). Then minimize this expected cost by differentiation.

For the second one: I assume you working with a standard normal random variable satisfying , and you try to find the pair to minimize the distance

Note the condition is equivalent to . Here since is a constant, once either one of the is fixed, the other one is automatically uniquely determined.

So here e.g. you may write as a function of . Now differentiating both sides with respect to , we can obtain . On the other hand, due to the minimizing requirement (the first order condition). Make use of these facts you can prove your desired result. (Of course with the symmetric property as well)