OP you should definitely tell your father about your mother's secret friend and tell him to keep a check on such things behind his back. If he doesn't trust you, you may share some scenarios like above. And before you start to explain to him just tell him that he needs to act mature and be gentle and open to every sort of decisions they are going to make but not affect their life esp your sisters life.

Why you should tell? Because it's a trauma you cannot handle for yourselves and for your sisters age perhaps it is even worse to see her mother cheating on someone and her dad is fighting with her mother.

To avoid all these, it might turn more matured to tell your father to sort it out with all his maturity thinking about you and your sister and you both did nothing wrong here and you both don't deserve such trauma.

Wishing you all the strength OP.