1.) yes
2.) Is it a man's duty on the battlefield to let the women do the work? Sanji doesn't know the power level of anyone that threatens his lady-friends. His takes it upon himself to protect them. If you see a man hit a woman, do you just let it happen? No, you intervene because as a man you feel like you should protect the girl. Why is it okay for a woman to push a man, but not for a man to push a woman? Men protect women when the odds scare them. As simple as that. Sure some women use that against us, but then some women respect that, and find themselves a man that treats her with the same respect she's giving him. Sanji holds himself accountable for the safety of women because on the battlefield, he's the man. It's not about them being weak in his mind, it's about his self-made duty. That's something he grew up learning. Something he learned probably from his father, who may actually be a king. It's not sexist, it's chivalry. This isn't about pairings, this is about Sanji being what many would call a man.
3.) Nami's fist would drive any man into obeying her (unless they're her enemy, because I'm not placing any money on Nami beating Doflamingo or anything). Sanji isn't dumb enough to play with Nami's trust. Kinda like how Robin squeezed Franky's balls. Women know how to get their man in line when they think they can play dumb.
1.) No.
2.) Congrats on comparing a manga where people actively fight to RL, when neither are remotely comparable. In a manga where everybody, man or woman, needs to be able to fight, Sanji doesn't allow Nami or Robin to be strong. It's far beyond chivalry, he interferes not because he is a man, but because on some level he views them as weak and incapable of defending themselves. They're strong too, but Sanji doesn't act like it. His "self-made duty", if that's what you believe it is, is really arrogance. It's not up to him to decide who's strong and who's weak. Luffy would never act the way he does, because he believes in the strength of is friends first. They're on Luffy's crew because he believes in them. Your argument
might have cut it before the time skip. The reason they had a 2 year hiatus is because they wanted to get stronger, including Nami and Robin. Sanji refuses to acknowledge that, and he's wrong for it.
3.) doesn't matter. One more time, there has been nothing to show that he can settle down. He's a horndog, he's the person most likely to end up alone.