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A woman CANNOT, repeat CANNOT initiate divorce under islamic law.

Saying it over and over makes it no more true. You need to cite your source.

She must REQUEST permission from an Islamic judge to decide if she has valid reason for the "khul"

In other words, she initiates the divorce much in the same way a woman here would file for divorce with the court.

Even when a husband beats her severely by breaking her bones, she still has to pay the husband to get rid of the tyrant.

Payment is up to the courts. A high payment is for a capricious divorce. A low or no payment is for a justified divorce such as your example. All of this is based on the premise of the husband not being insane, impotent, or other things, which would qualify the wife for a haqq-i-faskh, the equivalent of an annulment.

Census? Source? What source? YOU are not a source

Any source. Do you deny that there are at least a billion non-Catholic Christians in the world? Do you think these people believe canon to be law?

Again, "if a tradition did not meet the requirements for a sahih Hadith, it was not accepted into the family of sahih Hadith by Ishaq, Tabari, Bukhari, Dawud, or Muslim. Therefore, the standard Hadith collections are considered by traditional Muslim scholarship to be sahih. Thus, it is unjustified to use a weak Hadith defense when a sahih Hadith is used to present the actions and teachings of Muhammad."

"woman has to ransom herself to be free from her tyrannical husband; she can ransom herself from her husband for more than he gave her…29.10.32"
m11.13 (Ref: 8, p546)
Ref: 9, p192)

Again, the woman needs to ask permission of a judge and "ransom" herself with an expensive payment.

The woman CANNOT INITIATE divorce proceedings.
Maliki school—khul is a divorce by giving something in return.
Hanafi school—khul is end of marriage with the utterance of the word ‘khul’.
Shafii school—khul is a separation sought with something given in return and with the pronouncement of the word ‘khul’. Can be through mutual agreement or through the order of Kadi on payment by wife to the husband a certain amount not exceeding what was given to her as dower (mahr).
If the husband forces the ‘khul’ then the wife can keep the mahr.
Khul can be demanded only in extreme cases. Can’t be on flimsy grounds. (Although a husband can initiate divorce on no grounds)
Hadis “The women asking for separation and ‘khul’ are hypocrites”.
Khul is makruh ie it is disliked

Your arguments are the talking points of islamist apologists who are "uncomfortable" with the truths of their faith.

I've quoted sources. Your answer is to state that those sources are optional and open to interpretation. The above states otherwise, but I suspect you're well aware of this.

quran002.228
" Divorced women shall wait concerning themselves for three monthly periods. Nor is it lawful for them to hide what Allah Hath created in their wombs, if they have faith in Allah and the Last Day. And their husbands have the better right to take them back in that period, if they wish for reconciliation. And women shall have rights similar to the rights against them, according to what is equitable; but men have a degree (of advantage) over them. And Allah is Exalted in Power, Wise."

"but men have a degree (of advantage) over them." Clearly establishes men and women are NOT equal in islam...according to your quran.

quran004.034
Men are the protectors and maintainers of women, because Allah has given the one more (strength) than the other, and because they support them from their means. Therefore the righteous women are devoutly obedient, and guard in (the husband's) absence what Allah would have them guard.

quran 002.282 Woman's witness is only half of man's.

quran002.222

"They question thee (O Muhammad) concerning menstruation. Say: It is an illness,

You: "In other words, she initiates the divorce much in the same way a woman here would file for divorce with the court."

No. She needs to ask permission, pay ransom, etc.

The husband has no such limitations.