Hi Synergy,
I will try to answer your questions. But please bear in mind that I am not an expert in islam or islamic culture so I might be wrong.
1. Why woman cannot walk alone in public, but has to be accompanied by a male family member.
For me, this is more a question of a culture, because I think that it exists also in other cultures than in islamic countries, ie. in India. Why do I think that it is more a question of a culture? Because in India you have two main religions, hinduism which is prevalent and islam. BUT, whatever the religion, the social rules are similar, it is the family who finds husbands/wives to their children, women cannot wear skirts, I would say that in India rules are even stricter, for example in north african countries, as I know, a man can touch his wife in public, he can hold her hand, and so on, but they cannot kiss. In India it is impossible, a man cannot touch even his wife in public... But in India women do not have to cover their arms, and belly can be seen. Once I was in India with a friend of mine and one day we put on a skirt, and I can tell you, we were followed by a man for almost three hours. So it is a question of culture.
But in general, this rule serves to protect a woman, I would say. Maybe it also depends on a culture. I come from a country where people are very reserved, and men in my country are desperate because they would like so much to learn to flirt with a woman but they do not know how... most of them would not talk to a woman they do not know on the street and so on. Of course there are exceptions, but in general, it is like that. But when I moved to Paris I saw that it is much different, and as there are many people from arabic, Indian, african countries, I could see the difference, because they, in general, do not have any problem with communication with a woman. I would say that it is because in their countries the relations in a family are much warmer than in my country, or at least if I compare it with my family and my friends families. I have a friend (woman) who is from Maroc, she is very beautiful and lots of men ask her for her phone number on the street; I told her, it is incredible, how many men ask you to go on a date with them. She told me, this is nothing, you should have seen in Maroc, it is much worse, when you go on a street, unless you go with a man, men stop you on the street all the time...
Also, I was told, by friends who spend a lot of on time in India or Arabic countries, that, for example, it is dangerous sometimes for a woman only to make friends with a man, in case he falls in love with her and she does not want to marry him, he can become very fanatic after, and even pay money to someone to rape her, just to take revenge on her because she did not want him. Also men from these countries who live in Europe told me this.
So I think now you understand better. I do not know whether Islam (Koran or sunna) impose this rule, but if so, It must be because God knows the hot blodded nature of people there, although I am aware that the hot blooded nature is a cliche to a certain extent, still, there is a lot of truth in it. Because I come from a "cold blooded" country:)
2. This is also a reply why a women has to cover her body,if you see a women covered like that and compare it with a woman wearing normal clothes, which one is more sexy? Which one would flirt with if you were a man? Although many muslim women cheat, they cover their head and chest, but in such a way that they are still very beautiful and sexy...

As regards the way some women cover up their bodies, really from head to toe that you see only their eyes, this is, I was told be muslim men, exagerated, and Islam does not have such a rule, so it is again a culture thing.
In the beginning, there was no such rule, but then the Prophet Mohammed started to cover head of his wives, because as he was a very important person who got official and an official visits all the time, he wanted to protect his wives from stares and remarks from men who came to see him. Later on, this rule was extended to all muslim women.
3. The question of virgins in heavens... I do not think that there will be any sex life in heaven, as the paradise is, as Koran says, unimaginable for us, and it is to no avail to try to understand what it is like. But, I guess he wanted to motivate people to behave according to the precepts of islam, and beautiful eternal sex with a virgin is a good motivation, I would say. Personally, I like sex, but I do not give so much importance to it as men usually do, I guess I am not the only woman like that, so I guess Prophet used the sex promise only to men. Also, I am not sure, that in Prophets time, anyone would care about woman' s sexual pleasure, it is quite a recent thing, in fact. Did you know that in prophet' s time a woman had NO VALUE? It was normal to kill a newborn girl, because everyone wanted to have boys. Islam in fact improved a status of a woman, by giving her right to inherit I guess one third of the property of her husband, and also because he said that sex outside marriage is a sin. Because at that time, it was normal to sleep with a woman without marriage, man did not need to marry a woman in order to get the right to sleep with her. And of course he has forbidden to kill newborn girls.
4.To hide a woman in a house, and paint the windows so that no one can see her through a window? Islam does not say that. On the other hand, I always put the curtains, because, in my room, where I am always reading something at my desk by the window, because I hate people going by staring at me:)
5. Four wives for a man: Prophet suggests that the best is to have only one wife, but if a man is capable to marry more of them, he can, but there is one condition, he must give them the same amount of attention, material things, and so on. If one woman has a house, he has to buy a house to all of them, there must be 100% equality. It is quite a challenge... Also, Mohammed made wars, and in wars a lot of men died, so logically there were less men than women, and if it was not possible for a man to marry more women, all these widows would have to live alone. Another condition to polygamy is: a man can marry another woman, BUT his first wife must agree with it. Unfortunately, due to a high level of corruption in arabic countries, if a man pays some money to an administrative officer, second marriage is possible even without the assent of his first wife.
Anyway, at that time it was normal to marry more wives, and some men had even 20 wives or more. Prophet limited this number to 4 while stating that is is best to marry only one woman.
6. As regards the rape, I do not know what to think about it. You need 4 man witness, and the burden is on a woman to prove that she was violated; BUT if the burden was not on her, she might very easily accuse any men that he raped her and it would be up to him to prove that it' s not true. And the penalty is I guess a lapidation. SO like this a lot of innocent men might be lapidated. It is better to let live a guilty man that to kill an innocent man, isn' t it?The very sad thing is, that in many countries, not only in islamic countries, but in India, as I know, as well, if a woman is raped, it is a shame for all family, ane very often, the pressure of people around her is so high that finally she commits a suicide. But, in my country, (I come from central Europe) it is not easy for a woman neither, when she is raped and goes to the police, very often policemen treat her like a prostitute, without any compassion, and so on. Rape is a source of a great shame for a women and most of them hide the fact that they were raped. But I am sure, if women cover up their bodies and do not walk alone in the streets, there will be much less of rape cases. Unfortunately, very often, it is family members who rape them, in most of the cases, not only inislamic countries but everywhere.
7. In mosques women pray in a separate area in order to concentrate well on the prayers, because they might be distracted by sexual attraction to someone.
8. You live in a different world so of course it is difficult for you to accept all of this. But if you were born in a NORMAL islamic society, let' s not talk about extremes, maybe it wouldnot be such a big burden. As I was told, yes, a muslim women is dominated by a man, but she has a very good life, because their man take a very good care of them, and their are happy. And no, in muslim countries women should NOT be beaten by their husbands. There is one sourate in the Koran that talks about that, if a woman does not obey a man (for example, has found a lover, and so on), a man can do three things: talk with her about what she' s doing wrong, "separate their bedrooms" and, yes, beat her, but the the arab word used for that means something like "strike her very lightly" so that she realises she' s doing sth wrong, so the Koran does not give right to a man to beat up her wife. Unfortunately, it happens...all around the world. If someone says that a man can beat a women, he is exagerating and acting againts what the Coran teaches.
As far as I know, the New Testament says like men have to obey church, and women have to obey their husbands. They must really obey them but men must treat them well, it is just like islam, only in our western societies we have changed this rule, and a woman can dominate a man.
Do you know, in which European country women are most emancipated? In Germany. And a German female friend of mine told me that now German women complain that German men are kind of... feminized, which is not sexy at all, she complained:)
I hope I gave you satisfactory answers to your questions... But as you can see it is only my point of view.