How can one deal with a mother who physically and psychologically abuses her son and throws him out of the house, especially after he involuntarily pushed her in self-defense, and how can one show filial piety to her under these circumstances?

Harm caused to a mother through beating or verbal abuse does not negate her right to dutifulness. It is obligatory upon you to be dutiful to her, treat her with kindness, and obey her in what is good, so long as it does not harm you. There is no sin upon you for what occurred unintentionally. Allah is Forgiving to whoever rectifies their intention after a lapse or slip regarding their parents. So, be patient and persevere in being dutiful to her, for dutifulness to one's mother is among the deeds most beloved to Allah, and the swiftest in reward and blessing in sustenance and lifespan.