In what Arab states having multiple wives legal and where it is not? How this changed over time and what countries were pioneers in the reforms?
While the legal conventions are there, and some laws limit it to 4 wives, it is the religious convention from the Qur'an (the spelling I will use) where it is strictly spelled out how you can have multiple wives. In The Qur'an and multiple Wives
the Qur'an stipulates that a man is responsible for the maintenance of his wife or wives. If a man has more than one wife, he has to provide separate living accommodation for each of his wives. Multiple marriages are a heavy responsibility on the male. It is not a pleasure trip as some people may assume. Some even imagine all kinds of sexual exploits involving a man and his wives altogether. However, such activity is not permissible in Islam. A man must divide his time equally among his wives. He may, for example, spend one night with each wife on a rotating schedule. If a man cannot maintain justice in the treatment of his wives, the Qur'an stipulates that he is to have no more than one wife.
In my Muslim History class some of this went back to both Arabic culture around the time of Mohammed (the spelling I will use) and his numerous wives, basically taking from his example as The Prophet whatever he did was eventually discussed and passed down in religious law over time. Although changes were made basically looking at his example the Islamic Jurists issued opinions and edicts for centuries about every aspect of life and culture. From what we studied at the time there was not much change on this - the number of wives was small since my teach joked, only older men would afford multiple wives but they didn't have the energy for them.
Although it is rare for many Muslim men to have multiple wives, mostly you will see them only with the wealthy families in the oil-rich countries in the Middle East and even then you might not have someone with 4.
As to the countries many of them have changed in the past 100 years. From the Status of Women in Islam
A movement to improve the condition of women began emerging in the Ottoman Empire, including its Arab areas, late in the 19th century. First it was concerned principally with education. Gradually, the movement began to concern itself also with the social situation of women. The first and main targets of reform were polygamy and the ease of divorce for men. In 1926 Turkey adopted the Swiss law code which made polygamy illegal and gave equal divorce rights to women. In Jordan and Iraq, polygamy is restricted. In 1935 Reza Shah abolished the compulsory veiling of women in Iran. This new law marked the dateline of social change for Persian women. However, when Ayatollah Khomeini's regime came to power women were veiling themselves once again to demonstrate their loyalty to traditional Islam.
I could not find an overview of countries directly, but if you check the Gender Equality Index and look at the links they have by country you will get an overview. They seem up to date, even noting the decline of multiple wives in Saudi Arabia, the Country Overview will get you to the topmost listing.