Is a father's angry hand gesture considered backbiting or disobedience? And is telling my sister what happened out of distress considered backbiting or disobedience?

What you did was a sin and a transgression of the Sharia that necessitates immediate repentance. This is because it involved showing exasperation towards your father, which is a forbidden form of disobedience to parents, especially if the father became aware of it. Allah Almighty says: (So do not say to them [anything that indicates] displeasure and do not repel them but speak to them a noble word), and the verse indicates the obligation to be patient with them. And if the father did not know about it, it is also forbidden because it constitutes backbiting, and Allah knows the treachery of the eyes and what the breasts conceal. Therefore, you must hasten to sincere repentance.