"Being a man/woman has no effect on society outside of one's own personal sphere." Your opinion. Not the Qun's teaching. As you admit, "I believe" this, and "I do not believe" that. These are your beliefs, not the Qunari's. However, The Sten also informs us of the roles given to Qunari. For example, a baker cannot be a farmer. He is assigned his role as a baker and the Qun demands he fulfill his given role. The baker cannot claim to be a warrior and become one. Such an attempt would be met with reeducation. This is despite the fact the baker could be male. Tell me, would society be affected negatively if the baker became a warrior? No. Yet the Qunari do not allow it. Krem's case is even more complicated than the baker's. The Qunari would not accommodate Krem's wishes. The baker could fit the role of a warrior with training, but this does not matter to the Qunari. Krem's abilities would not matter to them either.
The Iron Bull has no need of Krem, but he does value each and every one of his men. He has gotten to know them. They had become a second family of sorts. He struggles with the decision to recall them or allow them to be killed, even at the cost of his own place within the Qun, which he also values greatly. One possible reason to lie to Krem is that he wants to boost morale or confidence within the team. Reassurance. Affirmation. Another reason is perhaps The Iron Bull is describing what he thinks the Qun should dictate. Or there is another option, which is that The Iron Bull was telling a half truth. Yes, there are aqun-athlok, but he never said what the Qunari do with them. He made his statement, Krem asked if they are treated like real men, The Iron Bull dodged the question and answered with his own personal opinion that they are real men and then reassured Krem.
That he willingly submitted to reeducation proves that he does not always follow the Qun. This means what he said about gender recognition within the Qun is unreliable. Your opinion that the Qun would recognize Krem as male is based solely on The Iron Bull's unreliable statements.
I think you are ignoring The Sten's words. There is no other interpretation. He said what he said. A female cannot be a male according to the Qun. Period.
Okay. Sure. My opinion. And potentially that of the Qun. Why would someone's decision to identify as man or woman affect society? I don't believe there's ever been a moment in game where someone of the Qun specifically states "You saying you're a man and acting like a man even though you were born a woman is tearing the fabric of society apart". And when you say "These are you beliefs, not the Qunari's" where are you getting that? Is the tome of the Ashkaari Koslun a rare codex we can pick up and read? Because if not, than all we can do is speculate to the best of our abilities given the limited nature of the evidence presented. That is primarily why I use the phrase "I believe", seeing as I have no way to confirm everything. And neither do you.
Krem is not Qunari, so your comparison of the baker isn't really applicable. However, on that note, the Qunari who is placed into the role of a baker (which is a woman's job in the Qun, implied by Solas discussing seeing a female Qunari baker put a pinch of sugar in the center of each loaf, as her own small rebellion (if you talk to him about his journeys into the Fade)) is placed there based on an evaluation of physical and mental traits. While this baker may wish to be a warrior, it has been determined that by said traits, this individual would not be suitable as a warrior. Thus, his being a warrior would be a detriment in that he would not be realizing his full potential in his contributions to his society. And this is just hypothetically going with the idea that a baker would be a male in Qunari society, to fit your argument. What must be remembered is that the Qun takes a different view on what harms a society. In our own society, we very much believe in letting a person self determine their fate. This does not appear to be the case in the Qunari society.
You are describing possibilities, and not facts. While these may be the case, I do not interpret his actions to equate your assertions. If the Iron Bull is describing what he thinks the Qun should be like, than that may play into something we do see in our own world very often - various degrees of religious fervor. It is not unthinkable that there would be different levels of devotion amongst the Qunari. And one simply being more devoted than another does not immediately equate 'bad religion'. I think if you made that argument, there would be many people of different faiths who would find fault with that line of thinking. And, as I said before, since Krem doesn't care much about the Qun, there would be little reason for the Iron Bull to lie/half truth about the Aqun-Athlok. Now if Krem had expressed an interest in game about joining the Qun, this might have more meaning. But he doesn't. So there's not really any reason for the Bull to pretend that his religion would accept Krem, if they didn't.
You keep calling the Iron Bull unreliable, but this is solely based on your interpretation of events. If he were truly uncertain of the Qun, he would not have submitted himself for re-education. His willingness to do so, even knowing what happened to those who underwent it, is indicative that he truly believed in the Qun. Additionally, we've only had close contact with two individuals of the Qunari faith, each of whom were quite different. But that difference does not immediately indicate that one is 'more correct' than the other. Until we've actually visited Par Vollen, this is something we will only be able to speculate about.
"A female cannot be a male" but Krem isn't a female under the Qun. Krem says he is a man. Krem does manly things. Therefore, he fulfills the role as set by the Qun. The Qun takes issues with women doing manly things. Krem isn't being a warrior woman. He is being a male warrior, which falls into the right category. If you watch the video I linked earlier, Sten says the female warden must either be lying about being a woman, or a warrior. In Krem's case, he isn't being a woman, he is being a male who fights.