All praise be to Allah and may His blessings and peace be on His last messenger, Muhammad,
As for women riding horses, I don’t know of an authentic report about horses in particular, however I find no difference between the horses and other beasts of burden, such as camels and donkeys, etc. I must also add that the default in matters of customs and transactions is the permissibility. Therefore, there is no need to cite a proof on the permissibility. The opposite is what is needed.
Here are some reports on women riding beasts of burden:
Al-Bukhari reported from Abi Hurairah that the Messenger of Allah (blessings and peace be upon him) said:
“نِسَاءُ قُرَيْشٍ خَيْرُ نِسَاءٍ رَكِبْنَ الْإِبِلَ أَحْنَاهُ عَلَى طِفْلٍ وَأَرْعَاهُ عَلَى زَوْجٍ فِي ذَاتِ يَدِهِ“
“The women of Quraish are the best women who rode camels, they are most kind with their children and most caring for their husband’s wealth.”
Muslim reported from Shuraih ibn Hani’ that ‘Aisha (Allah be pleased with her) rode a camel, and it was somewhat defiant, so she kept on beating it, and the Messenger of Allah (blessings and peace be upon him) said to her,
“Adhere to gentleness.””
As for li’aan, this is a legislation that predates the last message of Islam. The Old Testament does have this legislation as well, as in Numbers 5:11-31. The difference between that of the current versions of the Old Testament and Islam is that in Islam, the woman doesn’t drink bitter holy water mixed with dust from the floor of the tabernacle, that causes the curse; and will not have her head uncovered; nor will it be said to her that if she lies, her thigh will rot, and her belly will swell.
In li’aan, the woman’s word will be the final word; no matter how many times the husband makes an oath. In other words, his oaths will only allow him to deny the paternity of a child, he alleges, to be the fornicator’s and not his. His oaths shall not prove her guilty as long as she denies guilt.
As for why li’aan is the right of the husband, that is because a woman may get pregnant from the fornicator, and by attributing that son to the husband, he will not only have to tolerate the adultery of his wife, but also, the paternity of a child that belongs to the man who committed adultery with his wife. This means that this son will be also considered a brother for the husband’s daughters and mix with them freely, etc. It is also so because a wife may not have more than one husband, whereas, in what we know of the laws of all prophets, and the practice of many of them, a husband may have more than one wife. This means that a woman who finds her husband having intercourse with another woman may have simply seen him with another wife. It is also because of the undeniable difference in emotions between the two scenarios of a man finding his wife committing adultery with another man and the wife finding her husband in that condition. In fact, li’aan may be a way to help those men not take the law into their own hands, because if it is silence and acceptance of a child that is not his or getting four witnesses to see what he saw, many may not do either, and may very well attempt to kill both parties involved in the crime.
Finally, the woman who sees her husband committing adultery shall not be forced to stay with him. She may ask for talaq or khul’ and she will be granted one of the two.
Allah knows best.