Mayuka attempts to catch Aoba up on the war currently happening; he falls asleep on her shoulder during the lecture and a LOUD scolding ensues.
There is also the meeting with Dio in the hanger on military ettiquite.
In episode 5, Mayuka and another soldier "show Aoba round the Cygnus"- and by that, they mean making him clean the entire ship. Aoba is initially adamant in his refusal to do chores of any kind (he attempts to to escape while Mayuka clings onto his arm and tries to explain the situation to him), but he eventually relents. The entire sequence is hilarious- ending with Aoba being asked to help an injured and sleeping Lee with his urinal bottle. Both Aoba and Lee's reactions (the latter literally wakes up the second the order is given) are comical.
The plot of episode 6 centers around an Allience promotional video to demonstrate the Coupling System.
While Kleinbeck is explaining to the video crew the areas that are off-limits on the ship, Fromm suddenly interrupts to say how lucky the Cygnus's crew is to be able to serve under such a beautiful officer. In most anime, this would elicit a Luminescent Blush and/or an irritated reaction from the female in question. Here, Kleinbeck's only response is a deadpan "Huh" without any change in her facial expression. Everyone else in the room is shocked and a very awkward silence is present for a few seconds before Kleinbeck goes right back to her briefing without missing a beat.
During and after their fight Aoba and Dio call each other Weeper and Siscon. Here's hoping it becomes a Running Gag.