Is intention sufficient in place of saying "Amen" when supplicating, and will the supplication be accepted if the supplicant is sitting with his back to the Qiblah and his face towards his wife, who is saying Amen with her heart not with her words, and is it permissible for a man to lead his wife, her unmarried sister, or other female relatives in prayer?

Reciting supplications (Du'a) in congregation after prayers is not supported by any legal (Shar'i) evidence, and it is better to refrain from it. Saying "Ameen" to a supplication is only done verbally. It is recommended to face the Qiblah while supplicating because it is more likely to lead to acceptance. Congregational prayer (Salat al-Jama'ah) is obligatory for a capable man, and it is not permissible to abstain from it except for a legitimate (Shar'i) excuse. The acceptance of good deeds is a matter of the unseen (Ghayb), and what is required is to perform acts of obedience with sincerity to Allah and in accordance with the prophetic guidance, while fulfilling the pillars and conditions of the worship.