Being suky of others success, romance in this case, means he’s jealous of them if he had no one that could love him in return. So I can’t really accept your argument since it’s entirely obvious that Croire is gunning for him, Shion not so obvious at this point yet, and Rona is just uncertain. But he definitely has women who fell for him so being sulky of others is just weird I this case.

No offense taken, but that’s a problem between the transition from Japanese to English. “Mahou” is commonly translated as “magic” by translators. “Majutsu” can be translated as “magic” in English, too. So how to differentiate them if there are two different words in Japanese that mean the same in English? And there are actually a few novels where both terms are used to differentiate system variations.
To make it all worse, magic, sorcery, wizardry, magecraft and so on all basically symbolize a paranormal effect that can’t be explained by physical laws in English, but there’s no clear distinction between each of those words causing a constant re-definition by English writers how they use the different terms in their own works.
While your subjective view about sorcery might be true, no doubt about that, depending on the circumstances in which you encountered that word and its attached definition, someone else might attach a totally different connotation to it depending on their own background.
Actually magic/sorcery in this novel requires casting/incantations to take effect but that’s simply omitted due to the MC’s cheat power. Thus even going by your personal connotation of sorcery, the word itself should actually fit.
But even without explicitly agreeing with or rebutting your opinion, it remains true that it’s necessary to differentiate the two different terms somehow. And that’s how I decided to do it. I hope you can somehow live with it. 🙂

Because in Az and Liaris case, they share mutual feelings. Meanwhile Renya has no feelings for either Shion, Croire, Emil or Rona.

So he can sulk because he has nobody, he loves. While he has women that love him, it doesn’t matter since it isn’t mutual.

Yea, when they first arrived at Kukrika it was mentioned and once more after that. But basically I imagine it to be a lie detection spell, too.