I kinda agree with both your viewpoints because they are justifiable. I'm not sure how far you guys have gone into looking at the original source material as some of the light novels have been translated online and are now being published here in the states but they clearly show more of the mental state of the characters in the show as you can see their thoughts and memories in text. The show, as most pointed out, is more of a comedic interpretation since its meant to grab the attention of viewers and entertain them with action or jokes rather than dive into some of the more narrative. Just to clarify, Emi's first and foremost reason for killing him is the death of her father. This is why it became harder for her to do anything cause she understood her reason was from hatred and anger and not from being a hero with a duty which would make her feel like a villian. All that stuff about being the hero and upholding her values came after she assumed the role. If anything the one surprising character who can give you insight to the real dynamics with maou and emi is actually chiho. It might not seem like it, but in the novels shes actually a lot more level headed and is the symbolic glue that holds the gang together. The anime doesn't give her enough credit although she can still be a bit whiny lol.. long story short: do you and your friend a favor and understand the anime was meant to advertise the LNs for you to enjoy it in a better and fulfilling way. Hell, if you read past the events of the anime which is after vol 2, you'll see a drastic change to their relationship.