Utena was hell bend to be a prince herself, that is to play a male role in a relationship. Though Touga tried to change her mind. Later Akio did changed her mind for some time and she came back to be a girl for him. Then she return to playing role of prince for Anthy.
I think your analysis of Nanami's reasons to duel is spot on. She indeed done that out of jealousy and desire to win attention of her brother.
However not so much of other duelists.
Miki wanted Anthy because she reminded him of how his sister used to be. The kind of innocent pure girl who likes to play piano. His sister was like that but then she 'grew up' and started to be more concerned with appearance and sex. Miki did not like that about her and wanted to bring back his sister to how she was. Anthy did managed to convince Miki that she is just as innocent and pure like his sister was back then and that got Miki to dule Utena over the Rose Bride. Anthy of course did that consciously, knowing it will work on him.
In contrast I would say that it is Juri who wanted to protect, though not Anthy. She wanted to protect Utena from Anthy and Akio. She wanted to dissuade Utena from continuing to play a prince role, because she thought it is both futile and dangerous to Utena. As a girl like Anthy will never be satisfied with a female 'prince' and will always betray her for a male. Just as it happen in the final ark where Anthty backstabs Utena for Akio. I think Juri is the only real lesbian or bisexual in the cast and actually took interest in Utena there.
I think lots of people who call Akio a villain have this kind of Juri 'white knight' mindset. Many of them are not female though.
Akio is the most complex one here. He is the most experienced of them all, but also the most cynical and disillusioned with life in general. He used ot be able to believe in Anthy's innocence and good nature, but eventually stopped being able to.
However he fondly remembered times when he was still young and naive and could still believe in Anthy's innocent sacrifice for him. However since he no longer could believe in Anthy he needed to find some other girl who could be that kind of innocent Rose Bride.
For that reason he organized the duels and made these other guys duel over Anthy. He did intended for Utena to duel them all and win and then become a new rose bride for him.
He believed that if Utena can endure all these duels for the altruistic selfless cause of protecting another girl, she she would prove her selflessness and lack of any ulterior motives enough for him to like her. She eventually did proved that to him.
However to become his Rose Bride she had to abandon the very girl she fought so hard to protect. Eventually that same altruism that proven Utena to Akio, caused her to fight him instead to protect Anthy. That miscalculation on Akio's part caused him to lose them both eventually.
Quite unfortunate turn of events for Akio I would say.
That leaves even me to ponder if it is possible to find or achieve that same kind of perfectly innocent altruistic girl, some kind of more realistic approach is needed with women.