Assuming the Kou Empire, being a Fantasy Counterpart Culture of China, practices agnatic seniority inheritance (as opposed to primogeniture where either Hakuei or Hakuryuu would inherit after his death), the Emperor after the First Emperor's death is his younger brother. Okay. But why is Kouen, not Hakuryuu or Hakuei (assuming they allow females to inherit) not next in line for the throne?
I don't understand what you're confused about but Chinese dynasties did not follow agnatic seniority. In earlier dynasties, the usual tradition was for the eldest son to be appointed as Crown Prince and succeed as Emperor one day. In later dynasties, the practice changed and the Emperor appointed a Crown Prince out of all his sons. Sometimes he chose the son he favoured and at other times he chose the most suitable/accomplished. However, Chinese Emperors have also chosen to name their younger brothers as heirs before, feeling that they were more capable than their own sons. In Magi, the first Emperor either named his younger brother as the Second Emperor in a previous will or the Second Emperor seized power (which probably wasn't difficult to do as the remaining children of the First Emperor were young, and he had the support of Ren Gyokuen). For the Second Emperor, he named Ren Gyokuen as his heir (though that might be a lie she perpetuated) and thus, she succeeded to the throne even though the general royal tradition is to name either the eldest or the most accomplished son as the next Emperor, and Ren Kouen is both.
Why the hell do you think that bitch had her two older sons murdered in the coup in the first place? Because they were old enough potentially lay claim to the throne with strong support.
Okay. This has been bothering me since my first viewing of the anime, but why the hell didn't Alibaba bother to give money to Cassim and his little sister when he was living as a prince? They were very clearly poor and living in the slums prior to Alibaba being taken away by his father. So, why didn't Alibaba ever do anything about this? His father is painted as a good guy, so he would possibly understand. I mean, is this ever explained in the manga? It just seems really out of character and jerkish for Alibaba to completely forget about his adopted family.
Not sure if it showed up in the anime, but in the manga Alibaba was ticked off with how Cassim was talking about how they shouldn't be friends anymore since were totally different people and that Alibaba should just scram. Alibaba then proceeded to punch Cassim, angry that Cassim said all that. Seems to me that Alibaba just kind of threw himself into his duties afterwards, and didn't really see a way to fix things since he was having a hard enough time dealing with how he was ostracized in the palace, so things just kind of left off like that. Plus, Alibaba specifically stated that he felt as if the king was really distant too, so he probably didn't have enough confidence to ask of such a thing.
Are Kougyoku and Kouha twins? They look REALLY similar to me. I thought they were identical twins prior to realizing that Kouha was a boy. And I know Kougyoku is a bastard child, and I remember reading somewhere that Kouha is an illegitimate child too. So, did they have the same mother?
We don't get any hint of that yet; in the manga when there are people talking badly of Kouha they only mention that his mother went insane. I'm pretty sure that they are not twins (I think Kougyoku said that the only ones who ever noticed her were Judal and Kouen, no mention of Kouha which would be strange if they were full-blooded siblings). On a side note though, Kouha's mom is different from Kouen and Koumei's.