I GET THAT IT IS HORRENDOUS TO BE CHEATED ON - no you do not.
That moment of realisation never leaves you and whilst people do move on, you never forget. This is not breaking up with a boyfriend, this is destruction of trust, family, life, plans for the future, children etc etc.
I have moved on but my ability to trust both men and women is fundamentally altered. My protectiveness towards my DCS is much more than before, when you realise their DF or DM really does not make them a priority.
4 yrs down the line, my EX comes and sees the DCs in my house. if he did not then they would never see him. The OW gradually eroded their contact time, he allowed her to place all sorts of demands on his time that took priority over his DCS, his DCS are not allowed in his home - truly pathetic. So OP, she has no rights - do not let what has happened to mine happen to yours.
BUT do NOT contact her. I know from friends it drives her mad that I have maintained a dignified silence no matter what shit, and believe me she has done some serious crap, she has thrown in our direction.
The arrangement works for you, him and your DCs - whether he is babysitting etc etc no one elses business bar yours and his.
My sympathies.
Oh and anyone who thinks it is easy in this situation and tells you it is great that they are still together etc etc, wait till the next humdinger hits the fan - my EX has just split up with OW and all the hell that i creating!!!