Is it permissible for a single Muslim working in another country to authorize his father to buy the sacrificial animal (udhiyah) with his own money for himself and his family in his home country? And should the father refrain from (cutting) his hair and nails if this sacrifice is permissible?

If your expenses are independent of your father's, then a single sacrifice is not sufficient for both of you. You can authorize your father to perform the sacrifice on your behalf in your country, or give him money to sacrifice for himself and his household. The person who intends to sacrifice is the only one obligated not to take anything from his hair or nails. As for those who share in the reward of the sacrifice with him, they are not obligated to do so, according to the more preponderant opinion of scholars. This is because the ruling in the hadith is specific to the one who performs the sacrifice, and it has not been transmitted that the Prophet, peace and blessings be upon him, ordered his household to do so. As for analogizing the one on whose behalf the sacrifice is made to the one who performs the sacrifice, due to their sharing in the reward, this is an invalid analogy in light of the textual evidence.