Is it permissible for my father to perform tayammum for prayer while there is ritual impurity on his body, and to combine prayers, and to pray lying down, signaling with his eyes due to his inability to sit or face the Qiblah, given his health condition which prevents him from moving or going to the restroom?

Your father is obligated to pray according to his capacity, for prayer is not waived for a legally responsible person as long as his intellect is sound. If he finds someone to help him with Wudu (ablution), he should perform Wudu; otherwise, he should perform Tayammum (dry ablution). If he finds someone to help him sit, he should pray sitting; otherwise, he should pray in the position he is able to, due to the saying of the Prophet, peace and blessings be upon him: "Pray standing; if you are unable, then sitting; and if you are unable, then on your side." In a narration by An-Nasa'i: "If you are unable, then lying on your back." And he should gesture with his head, or else with his eye. If he cannot find someone to help him remove impurities, or if it is difficult for him, he should pray according to his condition while taking precautions. If he does not perceive what is exiting from him (e.g., urine, stool), he should perform Wudu or Tayammum after the time for prayer begins, and he is allowed to combine two prayers, either by advancing or delaying them, due to the excuse of illness.