1) It is haraam for women to be with non-mahrams or to be seen by them.
No, it isn't. Allah didn't create women as sex objects, nor did he create men as sexual monsters. By the way, see here for that analogy of women being compared to diamonds or pearls: http://mymuslimthoughts.blogspot.ca/2015/05/please-stop-comparing-muslim-women-to.html
Also, see here for the clothing of women in Islam: http://mymuslimthoughts.blogspot.ca/2015/05/trying-to-understand-womans-dress-in.html
And see here how culture views people's perceptions of Islam: http://mymuslimthoughts.blogspot.ca/2015/04/does-culture-influence-our-views-on.html
And see here for what lowering the gaze actually means: http://mymuslimthoughts.blogspot.ca/2015/02/what-does-it-mean-when-quran-says-to.html
Many salafis use the verse 33:53 to segregate women, to make sure they are hidden from men:
"O you who have believed, do not enter the houses of the Prophet except when you are permitted for a meal, without awaiting its readiness. But when you are invited, then enter; and when you have eaten, disperse without seeking to remain for conversation. Indeed, that [behavior] was troubling the Prophet, and he is shy of [dismissing] you. But Allah is not shy of the truth. And when you ask [his wives] for something, ask them from behind a partition. That is purer for your hearts and their hearts. And it is not [conceivable or lawful] for you to harm the Messenger of Allah or to marry his wives after him, ever. Indeed, that would be in the sight of Allah an enormity."
However people fail to realize that this verse is directed to the Prophet's wives. It speaks DIRECTLY TO AND ABOUT THEM. As we are about to see, the Quran has guidelines for the Prophet's wives ONLY.
2) Women should be at home
The verse used to say this is verse 33:33
"And abide in your houses and do not display yourselves as [was] the display of the former times of ignorance. And establish prayer and give zakah and obey Allah and His Messenger. Allah intends only to remove from you the impurity [of sin], O people of the [Prophet's] household, and to purify you with [extensive] purification."
But lets go back one verse to 33:32
"O wives of the Prophet, you are not like anyone among women. If you fear Allah , then do not be soft in speech [to men], lest he in whose heart is disease should covet, but speak with appropriate speech."
Once again, God in the Quran speaks to the WIVES OF THE PROPHET. 33:32 even says the wives of the Prophet Muhammed pbuh are not like other women. Other women here refers to regular Muslim women.
Some people say, even though it refers to the Prophet's wives, this verse refers to all women since the mothers of the believers are examples for all women. But then go back to verse 33:53. It forbids the Prophet's wives to re-marry. Now, according to the logic of those people, can muslim women even marry? Clearly muslim women of today are not married to the Prophet Muhammed pbuh! So that logic fails. And note that this logic is one of picking and choosing of what to follow in Islam, which is obviously haraam.
Keep in mind many salafi websites have that logic of women needing to be at home. They're clearly screwed up and misguided.
3) Democracy is shirk, biddah, and haraam
A salafi claim. Islam mentions no formal ways of governing. Which is why throughout Islamic history we have had different types of government.
4) Democratic laws are man made and against Allah's laws
No. The Quran and Sunnah mention have only a FEW LEGAL PARTS, while the rest is related to spirituality and stories. 90% of the laws made and passed in western countries are not against Islamic law.
One of the problems Muslim countries have is that they try to politicize everything that is in the Quran. They go beyond the legal aspects, and try to politicize family affairs, relations with non-muslims, relations with women, and so on. What is spiritual is meant to stay spiritual, what is contextual (meant for specific times and places (i.e. slavery)) is meant to remain in that time period.
5) Sex segregation is fardh (mandatory) in Islam
See point 1.
Here are some fatwas on how separating the genders is not a requirement in Islam:
Some videos by Dr Shabir Ally:
6) Women get half the inheritance
Firstly, I have written about how muslim women today are independent. See here: http://mymuslimthoughts.blogspot.ca/2015/02/what-does-it-mean-when-quran-says-men.html
Muslim women today are just as much important breadwinners as men. It is not the duty of women to stay at home, and cook and clean.
See this link that says women can get equal inheritance: http://www.mwlusa.org/topics/rights/inheritance.html
7) Women cannot travel alone
8) Wives are to obey their husbands
Comes from verse 4:34. It however refers to obedience to Allah. See the following:
9) Muslim men are allowed to marry more than one wife
See here: http://mymuslimthoughts.blogspot.ca/2015/02/why-are-people-so-misguided-on-polygamy.html