Is it obligatory for a son to apply the حدّ (prescribed punishment) for sodomy to his unchaste, unmarried father who engages in indecency, mistreats his family members, and sells their property to pursue his deviant lifestyle, especially since the son lives in a country that does not enforce the Sharia punishments? Or should he try to guide his father to repentance and continue to honor and respect him despite the difficulty of doing so?

Sodomy is a grave indecency, and its perpetrator deserves to be killed as a [legal] punishment (hadd). The passive partner is also to be killed if he is an adult, sane, and not coerced.

However, the implementation of punishments (hudud) is the responsibility of the ruler, not individuals.

Regarding your father, as a sincere and trustworthy advisor, you must counsel him and remind him of Allah, explain the ugliness of his act, and guide him to righteous company that will help him repent.

If he does not respond, seek assistance from those who hold influence with him, so they may take his hand and deter him from this abomination, while you remain committed to honoring and being kind to him, for the father's right is not forfeited.