A good start would be to look at Phyllis Schlafly...
No, not all women are part of the feminist movement. Feminism is a belief in and advocacy for gender equality, but individuals have different beliefs and may not align themselves with the feminist movement.
There are many different kinds of music and bands which have had an effect on the feminist movement. There are bands like Bikini Kill and the Riot Grrrl Movement.
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Consider Shakespear's Sister as a pioneering essay in feminist movement
It had undertones of Lesbianism and helped women to transition away from an oppressive patriarchy to Lesbianism.
Feminist movement in America was led (mainly) by Susan B. Anthony.
United Democratic Movement - Kenya - was created in 1999.
Several unfair reasons still exist that were not solved by the feminist movement. One of the most egregious is the disparity in pay between men and women.
It depends on what you regard as the "feminist movement". Some would consider that fighting for women's rights was a precursor to the main feminist movement of the 1970s. If this is the case, then the feminist movement arguably began with Dame Roma Mitchell whose influence led to the formation of the Women Law Students' Society, when she was not permitted to join the Law Students' Society because she was a woman. On 23 September 1965, Mitchell was made a Justice of the Supreme Court of South Australia in 1965, the first Australian woman to achieve this position. Pioneering the Australian women's rights movement, Mitchell was also the first woman in Australia to be a Queen's Counsel (1962).
Technically, all you have to do is identify as a feminist. At worst, you'd be a bad feminist (as opposed to not a real feminist) because feminism is a movement, not an organisation. To be a good feminist, you'd have to engage in women's rights advocacy or women's rights activism.
The most easier way to find information about the feminist movement is on the Internet. The information also can be find on magazines or books which are related to feminism.