The sentence that has correct pronoun usage is the second one - Either your supervisor or the restaurant owner must give their permission for you to have New Year's Eve off. You must use <em>their </em>because you don't know whether they are male or female.
d. Jones (2006) stated the results of the study directly correlate with the challenges...
Including the author's last name and the date between parenthesis is the best way to incorporate a citation directly within a sentence. To properly cite information inside of a paraphrased text, you have to include the author and the date of publication of his work.