The horse (Equus ferus caballus) is one of two extant subspecies of Equus ferus. It is an odd-toed ungulate mammal belonging to the taxonomic family Equidae.
Yes, it is permissible, and horse hair is tahir, so there will be no problem with that.
And Allah knows best.
Further explanation: In some cultures, dating back to ancient or mediaeval times (including parts of Europe/Britain as well as Greece), it was considered inappropriate or immodest for respectable women to ride horses in the normal way, especially if they were wearing long dresses that might not stay in place, so women were expected to ride side-saddle or, as Sayyed mentions, in a litter atop the animal. (I am not saying that no women ever rode horses the normal way, just that this was considered more mannerly.) It seems that women riding horses the normal way may have been seen as suggestive, although, nowadays, in much of the world, there is no stigma attached to it.
If they rode bare horses or regularly saddled horses during that time, it could be taken as a sign of them being treated harshly by the enemy.
Muslim women used to sit on Hawdaj which is like a seat put on the animal and not directly on the animal. Sitting directly on the animal or on saddles is Makrouh for women.
Yes. This is mentioned in some books of history.