Is the following Hadith authentically established: "There is no prayer for a man in his house without an excuse," and what is the expiation for the sin of transmitting it if it is not authentic?

The meaning you mentioned to your father is correct, and there is a hadith to that effect: "Whoever hears the call [to prayer] and does not come, there is no prayer for him except for one with an excuse." This was narrated by Ibn Majah, Al-Daraqutni, Ibn Hibban, and Al-Hakim, and its chain of narration meets the conditions of Muslim, though some preferred to consider it mawquf (a saying of a companion rather than the Prophet).