Having spent some time in places where chivalry is still very common I've personally found that societies which still rely on chivalry are also places where women on average have less power.
It really makes sense. What do men get out of chivalry? Obviously you're expecting something from the woman, she has to uphold some part of the bargain. Otherwise are women just better than men? Are they more deserving of special treatment?
Actually, biologically speaking, women are way more valuable than men. Or perhaps not the women and men as such, but women's eggs and ability to bear a child as opposed to men's sperm. Each egg is invaluable compared to each sperm.
To take a small example: If you have a tribe with a hundred men and a hundred women, but only food enough for 120 people, who must go without food and starve to death? In all rational culture, we would probably see something like, say, 20 women and 60 men starve to death. Why not fifty-fifty? Because a lack of women would be a population bottleneck far more than a lack of men would be, as each man, needing only minutes to procreate, could father children with multiple women (while the opposite is rarer, each woman needs about a year to bring about one child). So each individual woman is more valuable for the survival of the tribe than each individual man.
Of course, as women then have higher inherent value than men, families and tribes are less likely to let women go and do whatever they want than they are men, simply because if they lose a woman, it could be a small tragedy, and if they lose a man - well, he can always be replaced.
But this quickly leads to men being able to do stuff that gives them recognition as men, while women aren't able to do much. This again leads to the idea that daughters are only valuable to their parents when they can be traded to a man in return for allegiance, favors or goods. Which again continues with the woman - now being the man's wife (hopefully...) - is only good for making children and taking care of the house. And not to forget: No man wants to raise another man's offspring, and the only sure way of knowing that a child is his is to make sure that the woman doesn't have sex with any other man. Hence the value of virginity and of trying to avoid women having contact with unrelated men.
And so it continues.
Chivalry then, actually comes in at the beginning of this story as well, as rational rules of conduct: Since women are more valuable than men, men should go out of their way to protect and provide for the women. Of course, nobody spends time and resources or risks their health or life for something that has no value, and since women didn't have any value except as virgins until they become some guy's wife and produce children, women who broke these rules would find themselves without the protection from chivalry. And a lone woman without anyone to help or protect her doesn't stand much of a chance in this world. Not that this 'being alone and vulnerable' wouldn't be true for all/most men as well: Being exiled from your tribe/country has never been an easy way out.
So to answer your questions:
In theory, from chivalry, men get wifes who will bear their children, and their children only. Women are not generally "better" than men, nor are men generally "better" than women, though - with men being statistically stronger than women - in more primitive societies, men would be more useful at many hard physical tasks (not that women didn't have to contribute to such tasks anyway, especially if the family/group was poor). However, men have always been more expendable than women, so they're the ones most - all - societies use for waging wars. Women are for the most part not put in the most dangerous situations, specifically because their eggs are more valuable than mens sperm.
And finally, is chivalry useful today? Both yes and no? Some points:
- It's a generally nice thing, and it would be really good if all people could be courteous, polite and helpful to one another.
- The reasons for why chivalry exists are not eradicated. They are simply hidden behind all the layers of modern civilization, where we have the rule of law, where the state holds the monopoly on violence and where our technology and resources allow us so many greater freedoms and opportunities then before. However, as soon as aspects of the modern civilization diminish (in a war torn country, in a crime-ridden back street, far away from other people, etc.), the reasons for chivalry will surface again. This is something to have in mind.
- Chivalry does not help in seducing a woman. It worked in older times, because then it wasn't the woman one needed to seduce, but the woman's father, who wanted to know his valuable daughter would marry a good man. Chivalry in itself is not a point of initial attraction.*
- However, being chivalrous (to anyone), shows that one is a nice person, which will at least make other people (both men and women) think of you as a nice person.* And in all social interactions, that is always the first step.
* True, a select few women will of course get offended by chivalry. But hey, you can always find some stupid human who gets offended by something.