When did the rationale about abortion change from eugenics to a woman's right to choose?
It didn't, those are two separate things, a woman (or more likely, a couple) has the right to choose to terminate a pregnancy if after prenatal screening they find a big probability that the baby will be born with certain unwanted conditions. I can't speak for all countries obviously, but I know for a fact in my country you can do it up until the 25th week or in some cases without even a time limit.
Is it right for China to fine tune its population through abortion? What can be done about the surplus of Chinese men due to preference among Chinese to abort baby girls?
That being said, I have no idea what the law in China is about this, but I think in most countries they don't allow this, abortion is usually allowed until the 1st trimester without questioning the reason and after that, it's allowed on certain conditions.
What can be done about the surplus... not much really, they have to realize on their own how sexism and misogyny ended up being bad for their society and live with their decisions now.
Should a man have any rights in the abortion decision?
I don't see the reason or how that could even practically happen. Like let's say a woman wants an abortion and a man doesn't, what's going to happen? Take the case to court? Or the opposite, a man forcing a woman to have an abortion. Just imagine how that would play out.