With regards to fitting this in between EIP and Sozin's Comet, my headcanon timeline places EIP about six weeks before the day of Sozin's Comet, so that leaves plenty of time for other adventures. For this story, I see Zuko as having made the suggestion that started it the day after EIP and the story itself starting two days after EIP.
As for Zuko's relaxed reaction to being captured by the Earth Kingdom, he's thinking logically. He knows they've recognized him as the Firelord's son, which means that they likely think he has value as a political bargaining chip. Even if they aren't thinking of that, they're unlikely to kill someone who could provide valuable information about the Fire Nation's plans. And once he tells them that he's changed sides and is teaching Aang, he has a further layer of insulation from serious harm.
In contrast, my headcanon for Zuko's status in the Fire Nation post-DOBS is that he's wanted alive or dead and the primary incentive for keeping him alive is that the reward for turning him in alive is substantially higher than the reward for his corpse. Zuko also knows that the only reason he's wanted alive is because Ozai wants to kill him himself.
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As for Zuko's relaxed reaction to being captured by the Earth Kingdom, he's thinking logically. He knows they've recognized him as the Firelord's son, which means that they likely think he has value as a political bargaining chip. Even if they aren't thinking of that, they're unlikely to kill someone who could provide valuable information about the Fire Nation's plans. And once he tells them that he's changed sides and is teaching Aang, he has a further layer of insulation from serious harm.
In contrast, my headcanon for Zuko's status in the Fire Nation post-DOBS is that he's wanted alive or dead and the primary incentive for keeping him alive is that the reward for turning him in alive is substantially higher than the reward for his corpse. Zuko also knows that the only reason he's wanted alive is because Ozai wants to kill him himself.