Is it permissible for someone who has donated a horse for the sake of Allah to enter it into competitions, and what is the ruling on the prize if it wins, and does its offspring also remain for the sake of Allah?

There is no harm in entering horses into competitions for their training and the training of the competitors, as the Prophet, peace and blessings be upon him, raced horses. It is a condition that this does not harm the horses, based on his saying, peace and blessings be upon him: "There should be no harm nor reciprocating harm." If the horses win a prize or give birth, all of that is for the sake of Allah, because the yield of an endowment is تابع (dependent) on it and is spent in its designated ways.