Is it permissible to give a poor mother from Zakat money when one is able to give her from other than Zakat, and is it permissible to leave the father with whom the questioner lives due to psychological discomfort and frequent disagreements?

It is not permissible to give zakat to your mother or grandmother if you are obligated to provide for them. You must treat your father with kindness, never harming him with words or deeds, and be patient with whatever emanates from him, while striving to be dutiful to him and live harmoniously with him.