1. I don't think Mikasa is a psychopath. If someone saved your life as a young child from robbers, and put you in a position where you had to kill at a young age, I find it reasonable to devote your life to protecting them given you have the skill (which she does).
2. It's science fiction... I'm really not sure why you're so annoyed by all of these things. Do you get annoyed by Gundam Seed because everyone knows that hyper-particle beams of that size clearly need at least 4.2 seconds to power up and the beam should be red, not blue?
3. I still don't understand what sort of "message" you think is trying to be pushed through here, as well as why you find it in poor taste considering most adult anime does have at least some sort of message to some degree. Which is why I recommended Pokemon... Or maybe teletubbies. Although the Ukrainian government believes teletubbies, too, is pushing its own hidden agenda and has banned it so perhaps maybe just Pokemon. Or those chibi specials.