All three of your questions stem from Act 1, sc. 1. Simply put, Theseus is anxious for the new moon because that is when he and Hippolyta will get married. This is explained in the first 6 lines of the play. It also appears that Theseus is more anxious to get married than is Hippolyta, but considering that Theseus defeated Hippolyta in battle, that's not that surprising. In lines 12-16, Theseus tells Philostrate to get people in Athens stirred up, or excited, about the upcoming wedding. He wants there to be a joyous and excitement-filled atmosphere; a holiday-like attitude among the people of Athens. As to why Egeus prefers Demetrius to Lysander, there is less clarity. It is probably because Egeus chose Demetrius and he didn't choose Lysander. Clearly, Egeus is old-school in that he feels that he, as a parent, knows best who is a suitable mate for his daughter. Lysander has wooed and charmed Hermia with singing at her window, small gifts and treats, and poetry (lines 29-35) which is new-school. This alone is enough to turn Egeus against Lysander because he, Egeus, wants to choose his daughter's husband. We are told that Demetrius and Lysander come from the same social strata so that is not a reason. I think the choice of Demetrius initially was just chance and little more.