Fisher Information and Wilson Confidence Interval. B+D only.

Note: I have the answers for a+c I am looking for b+d only.

a) is 


c) is


for d) it has to be calculated using this theorem to get the aymptotic distribution.



This is what the expected final answer should be


It would be much appreciated if you could be thorough in the explanation, especially tying the Asymtotic normality of the MLE into everything and how it relates to the Wilson confidence interval. Thank you!