Are the profits of a training center permissible (halal) or forbidden (haram) if it relies in its advertising on designs made with pirated software? And is it necessary to replace the old designs after repentance and purchasing the software?

It is not permissible to use copied programs without the permission of their owners. However, if your husband was unaware of the ruling, there is no sin upon him. He may use the previous designs, but it is not permissible for him to use these programs in the future except with the permission of their owners or by purchasing the original versions. The profit gained from the training courses is lawful.