Is it permissible to use another's stud without his permission, on the pretext that taking a fee is forbidden, and is permission a legitimate requirement? What is the ruling on using the owner's horses for breeding without his permission? And what is the owner's right in this case?

It is forbidden to take a fee for studding a male animal due to the Prophet’s (peace be upon him) prohibition of the earnings of a male animal. It is not permissible to use another person's male animal except with their permission. Whoever uses it without the owner's permission is an oppressor. The offspring belong to him, but if this leads to a defect in the male animal, then he is liable for compensation. If it does not lead to a defect, then there is nothing upon him except repentance for encroaching upon another's right.