Is it permissible for a person praying at home, while the congregational prayer is being established in the mosque, to say the prescribed post-prayer supplications (adhkar) after his prayer, even though the Imam is reciting the Quran, or is it obligatory for him to listen attentively?

The questioner is not obligated to listen to the Imam's recitation from the mosque while she is in her home, whether she is engaged in the post-prayer supplications (mu’aqqibat) or not. Listening to the Qur’an is only obligatory during prayer. There is no harm in her engaging in the post-prayer supplications.