There is a secular approach to issues, and there is a Torah approach to issues. The secular approach tends to be that if there is an issue, then something is wrong and has to go. In this case, if there is a "war of the sexes", then the idea of "sexes" needs to go to end the war. The Torah approach however is that if something in Hashem's perfect creation is not working properly, it's because we are doing it wrong and need to figure out how to do it right. In this case, we are looking at it too superficially, and the Torah's response is go deeper, and states unashamedly that there is no greater hatred of women than to tell her that her femininity is bad/wrong and must go out the window (same goes for men).
In this answer I hope to bring together the words of many great Rabbis who have taught the inner dimension of male and female, from the kabbalah, and chassidus. For further reading/listening from Rabbonim who say it infinitely better than I ever could, I recommend Rabbi Tatz's Living Inspired, Rabbi Yitchok Ginsburg's discourses, and anything you can find by Rabbi Moshe Miller A"H on the topic. I hope to cover all the points in the question and I note I will not be adding any words or ideas of my own, and in particular I am basing this answer on this lecture "The Kabbalah of Male and Female" by Rabbi Manis Friedman, who is a master of the topic and a student of the Lubavitcher Rebbe, who also covered much of this in his sichot (which you can find on Chabad.org) and I strongly recommend listening to all his talks on this topic. Finally, may I strongly recommend "The Story of Our Lives" by Yaakov Klein, which explores the connection between femininity and Malchut, which will be heavily connected to the themes in this answer of women being the purpose, the end goal, of creation.
I apologise for the length, but these ideas shouldn't be too simplified (which I apologise that I will inevitably end up doing even at such great length... please do take my advice for further reading!).
The Torah explains that God created "Adam", male and female, and said it was "very good". Then a few verses later, he said "it's not good", and he took a part of Adam and made Chava. This requires analysis. Hashem does not change His mind or require adjustments in His plan as He goes along, so what's going on here?
We have to come to an understanding of why Hashem made both male and female, and why female was necessary if there was already male (Hashem doesn't make anything redundant!). Hashem initially made a man, who had a female dimension. He then split off a woman (who has a male dimension). He then commanded them to get back together, something possible because they were once one (there's nothing new under the sun).
Hashem created the world, and a physical universe needs certain things. One of the things that a physical universe needs is a male. When He made that, He said that it was "very good" because it is. A physical, finite universe has what it needs - a male. This is stage 1. Things are not good yet for stage 2. What is stage 2? Stage 2 is to take this finite, imperfect universe, and make it holy, through commandments, Godliness and goodness, so that it should become a dwelling place for Hashem Himself. So for this part of the plan man alone is not good. Not "he is not good alone", it will not be good if he is alone. To understand why this is, we must first discuss the masculine attribute that Hashem created, then the feminine:
The masculine is a process. The finite universe needs to become receptive to good and an infinite God; it is far from complete. Masculine is a step towards a goal - foreskin, the wheat for matza, the wax for shabbat candles, the leather for tefilin - it's all a process that needs to be transformed into Godliness, and when Moshiach comes the world will gain eternity when it becomes one with Hashem. Man is created as part of this side of things - a mortal finite creature, incomplete and imperfect, with a need to transform and become Godly. This is why men have a need to fix and pursue and find purpose and mission. This is why in the kabbalah, the male is linked with time. Time is a process where things "happen", and that's the male (interestingly, this is why there are 14 mitzvot women are exempt from, because they are time bound).
However, Hashem noted that this masculine energy will not guarantee the world reaches its purpose - a holy, complete world. "It" - the plan - is not good. Where will the man get the momentum, the motivation, the clarity of vision, to complete the plan?
A mashal is given, where a group of scientists get together to plan a mission to go explore another star. This kind of mission will take many millenia to see through, so they have to send a whole community, a generational starship. They discuss the issues of the cleanliness, and morale. The issues of wear and tear and degredation. Then one scientist brings up a really big problem. He says "what about the great great grandchildren of the original crew, when they wake up one day and say 'who said we are heading for a star? Has anyone actually seen earth?'. The control room they are not allowed to go into, because only the high priest is allowed to go there, they say 'enough of this, anyone can go in'.... The scientists hit the nail on the head about this same issue we are discussing. How are we supposed to guarantee they won't just say "forget the process, things are fine the way they are, let's make the best of it...".
Hashem made woman to "help" with this.
So what is a woman? When God created man, it was because creation needed a man. But woman wasn't created as a means to an end. Woman is created because He needed her. Female is an end in itself.
Another way to put it, Hashem invested Godliness in man, to be processed and refined and brought to completion. Hashem invested Godliness in a woman perfectly as it is supposed to be, and the goal is to maintain it, nurture it and share it. This is what is meant by the phrase "to work it and to keep it" (Bereshit 2:15) - it is referring to the male and female aspects.
Another way to put it. It is the nature of man to be "aggressive" and the female to be "passive". Let's not understand this superficially or we risk re-igniting the war of the sexes! As I said at the start, let's go deeper. "Aggressive" means, there is something I don't have yet, I go out and seek it. Acquisitive. "Passive", which is often just as much (if not more) work, strength, energy, is the art of defending, maintaining and nurturing what's already precious. A lioness defending her cubs is not weak. This is why the child belongs to the mother (if she is Jewish, he/she is Jewish, if not, not). The innocent child is already good, a Divine gift of Godliness, and the woman is the mother, she is the child, and she is the home. The father is concerned, must provide but it's not the same.
This is also why the 6 days of the week are "masculine", where the work is done. Shabbat is the Queen, it is the destination, the reason and purpose we work on those 6 days. It is also why women make the bracha "who made me according to His will". Men cannot make this bracha because they aren't made a finished product like women are. They must process themselves and the world to become what it needs to be: feminine!
When the process gets side-tracked, confused and direction is lost, women have always saved the day. When in Egypt the men became Egyptianized, the women brought them back and enabled the redemption to take place. When the men became Hellenised by the Greeks, the women brought the nation back on track. No women participated in the Golden Calf, and it was the women who donated to the subsequent Mishkan.
It should be noted, when the world is full of dragons, the dragon slayers get all the attention. However, why would anyone slay a dragon if there was no one to save from the dragon? Why would soldiers go out to war if there wasn't someone precious to protect? Why would a doctor qualify if there was no "lay man" to heal? Without women, men have no purpose (perhaps this is why men can get very touchy about feminism, whose main line is "women need men like a fish needs a bicycle" - when a man hears that, he hears that he has no purpose to exist...). So who is the ikar? The masculine or the feminine?
In Kabbalah, there are two sides, the side of kindness and the side of severity. Which one is male and which is female? It might surprise many to learn that Kabbalah describes the masculine as Chesed, and the feminine as Gevura. Chesed is indiscriminate, everyone can have everything, very open. Gevura, however, is discriminate. It judges, it weighs, it sees value and this power of being able to discern leads to the intuitive ability to see the truth that women tend to have. This is the reason women are not allowed to be judges. They already know who is telling the truth before any case has been made! Men have to go through the process, but women, before they can even get to the process, know who is guilty. For this reason, they are disqualified in the same way a prophet is disqualified! A case must have a process, circumventing it means there is no chance for appeal. Rabbi Meir would consult his wife whenever he would rule on a case!
As we move closer to Moshiach, there are fewer dragons to slay, and the masculine trait becomes less and less, and the feminine trait becomes more and more dominant. What do you do with the good things in life? Who knows how to take care of the holy, Godly things in life? The feminine. This feminine trait within man has to become more dominant - they must seek out that which has always been Godly and good, and nurture it. As Moshiach comes, the wife becomes "the crown of the husband". Hashem didn't put men into a world that is purely dark, and purely evil, for us to struggle with on our own. He placed the goodness here as well - hidden for 5000 years - and the time has come to notice it, to cherish it, to savour the Godliness and goodness that Hashem placed into the feminine all along.
Conclusions:
Hashem distinguished masculine and feminine by giving them different energies. They get ta'anug, satisfaction and pleasure from different things. The masculine gets pleasure and satisfaction from fixing the world, from starting out finite and becoming Godly. The feminine gets pleasure from nurturing and savouring that which has always been Godly ever since Hashem said "I will make you a help mate".
As a result, many differences arise between the nature of male and female, but this difference is stemming in a very deep place in the psyche. The way this arises to the more superficial aspects of our psyche can lead to some overlap. Hashem also gave us a share of each others' energies - there is a feminine aspect within a man that he can relate to, and visa versa. However, the differences are deep, and important, and will lead to measurable differences in the personalities and character traits of men and women (on a statistical basis), and the sages captured many of these differences in their writings in the Talmud (some of which are mentioned by Dr. Maurice Mizrachi in his answer, for example). We should be aware of them, as good personal advice to which each person will relate to in a personal way, and we should certainly not mock each other for our differences.
This knowledge is true and therefore binding, it is from Hashem's Blueprints themselves, and if we ignore it or start mixing things up between masculine and feminine, we will end up dissatisfied, confused and depressed, especially if the feminine were to try to become more masculine right around the time the world is becoming more feminine! The Kabbalah is Hashem's deepest wisdom. The fact that it paints these things more deeply and extensively than the Talmud may be due to the fact that the Sages were not allowed to teach the secret part of Torah back then, and the revelation of the meaning of every lesson in the Torah has been a hard-earned slow and methodical process throughout our history of maturing and deepening our understanding... but it is the deepest truth on the matter, an illumination of Hashem's secrets and we should be glad to know our true natures, our inner selves, and where our energy is coming from, and use it, take pleasure in it, and realise that both the masculine and feminine are different expressions of One Godliness. We should use our talents, stay true to our Godly natures, and men and women should feel how valid and vital they are to Hashem's purposes B'EH, and feel empowered by it.
Women are the real treasure of the world, the purpose of creation and the feminine millennium, where we move into our eternal Home with Hashem, and enjoy the hard work of 3000 years, is right round the corner.
