Is it permissible to designate a daily litany for listening to, studying, and writing one of the Beautiful Names of Allah, or is that considered an innovation (bid‘ah) that was not transmitted from the Prophet (peace and blessings of Allah be upon him)? And how did the Prophet teach the Companions the Beautiful Names of Allah?

There is no objection to dedicating one day to studying one of Allah's Beautiful Names, and another day to another Name, as this is a varied method of learning. As for campaigns of seeking forgiveness (Istighfar) and the like, please refer to the mentioned fatwas. We have not come across the specific method by which the Prophet, peace and blessings be upon him, taught his Companions the Beautiful Names of Allah. For further benefit, you may refer to the article "From the Guidance of the Prophet, peace and blessings be upon him, in Education."