Not exactly on the quotes and definitely not on the Bible talking bad about women.
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Genesis 2: 20-25 is
"20 The man gave names to all cattle, and to the birds of the air, and to every animal of the field; but for the man F8 there was not found a helper as his partner. 21 So the Lord God caused a deep sleep to fall upon the man, and he slept; then he took one of his ribs and closed up its place with flesh. 22 And the rib that the Lord God had taken from the man he made into a woman and brought her to the man. 23 Then the man said, "This at last is bone of my bones and flesh of my flesh; this one shall be called Woman, for out of Man this one was taken." 24 Therefore a man leaves his father and his mother and clings to his wife, and they become one flesh."
There is nothing about a "rib" in the literal Hebrew, rather the word is "side". The "rib" translation was a result of rabbinical teaching in the late Middle Ages. In the King James version it is "help meet", which in modern English is partner. Also, 'adam in this passage was not a name, but rather a term meaning literally "groundling" (of the earth) and at this point in the story if taken in context of chapter 1, was "male and female" (see 1:26-27 "Then God said, 'Let us make humankind in our image, according to our likeness; and let them have dominion over the fish of the sea, and over the birds of the air, and over the cattle, and over all the wild animals of the earth, and over every creeping thing that creeps upon the earth.' 27 So God created humankind in his image, in the image of God he created them; male and female he created them.")
There was a blame game when man and woman were confronted with sin (again, neither had been named at this point). However, man doesn't blame woman---he blames God. 2: 12 "The man said, 'The woman whom you gave to be with me, she gave me fruit from the tree, and I ate.'" Woman blames the serpent. 2:13 "Then the Lord God said to the woman, 'What is this that you have done?' The woman said, 'The serpent tricked me, and I ate.'"
There were consequences to woman's actions; however, nothing in the text expands these consequences to all women. Nor are is the second consequence "punishment", rather it is descriptive and not prescriptive. Genesis 2:16 "To the woman he said, 'I will greatly increase your pangs in childbearing; in pain you shall bring forth children, yet your desire shall be for your husband, and he shall rule over you.'" As a result of her teshqua (here translated "desire" though no one really knows what it means) is for her husband, he (man) will rule over her. Again, nothing expands this to all woman or to all men.
Man is punished, but on severity--that's subjective. 2:17-19 "And to the man he said, 'Because you have listened to the voice of your wife, and have eaten of the tree about which I commanded you, "You shall not eat of it,' cursed is the ground because of you; in toil you shall eat of it all the days of your life; 18 thorns and thistles it shall bring forth for you; and you shall eat the plants of the field. 19 By the sweat of your face you shall eat bread until you return to the ground, for out of it you were taken; you are dust, and to dust you shall return.'" He did "harken" but he also ate. He'd already been working for a living, after all 'adam, male and female, had been created to have dominion over the earth. Now, however, the earth would be cursed and God wasn't going to supply the food---man would have cultivate. Whether that is more of less severe is again up to discussion.
Lamech is one of a long line of polygamous Biblical figures. However, as Genesis 2 (see above) shows, God's intention of one man leaving his family to joing with his wife (kind of hard to live your family and join with your wife if you're already married.) Further as the account of Lamech shows, the Bible presents polygamy in negative terms with very negative consequences. Lamech is not shown to be a positive figure, rather he immulates Cain in committing murder and completely misses the point that what Cain had done was wrong. 4:19-24 "Lamech took two wives; the name of the one was Adah, and the name of the other Zillah. 20 Adah bore Jabal; he was the ancestor of those who live in tents and have livestock. 21 His brother's name was Jubal; he was the ancestor of all those who play the lyre and pipe. 22 Zillah bore Tubal-cain, who made all kinds of bronze and iron tools. The sister of Tubal-cain was Naamah. 23 Lamech said to his wives: 'Adah and Zillah, hear my voice; you wives of Lamech, listen to what I say: I have killed a man for wounding me, a young man for striking me. 24 If Cain is avenged sevenfold, truly Lamech seventy-sevenfold.'" As far as God approving such marriages, keep in mind Lamech was completely disregarding how God had defined marriage, and that God specificaly said that kings were only to have 1 wife---an ordinance David and Solomon disobeyed. Remember too, the account of Hannah's pain that resulted form polygamy and the bitter rivalry of Leah and Rachel. The Bible provides very vivid warnings against polygamy and is consequences.
Now about the Bible and women:
Yes, the Bible was written to flawed people through flawed people. However, the Bible shows that women are created in God's image (see above). You've only mentioned OT, but in the NT Paul writes that we (including women) are to be conformed to the image of Christ. The Bible gives us moving accounts of rape victims (Dinah and Tamar-David's daughter), the sorrow of infertilty (Rachel, Hannah); the frustration of having men think they should make decisions for women (the woman of Geshua).The Bible shows that women are to be heirs--and in the NT co-heirs with Christ. The Bible shows that women are to fully participate in faith, in both the OT NT. Women are portrayed as intelligent, creative thinkers (Tamar, the SyroPhencion woman). They are portrayed as leaders (Deborah, Lydia, Pheobe). As women of faith (Sarah, Eve, Dorcas, Mary Magdalene, Rahab). As teachers (Priscilla, Huldah). As evangelists (Mary Magdalene, woman at the well). Warriors (Jael, Deborah, the women at the wall.)
Of course, the Bible also has patriarchal themes; it reflects the assumptions of a patriarchal society, while challenging those assumptions.