I am afraid that there is no longer any such concept as " the custodial parent"
You both have full Parental responsibility even if the children in fact live with you the majority of the time.
The law says that children are entitled to spend time with both parents and the courts will enforce this - and when they are with their father he has complete control over what they do and where they go PROVIDED that he keeps them safe.
Yes it would have been better not to introduce a new partner until they have been together for at least a few months but that has now been done and cannot be undone.
Yes their lifestyle may not be what you like - but again I am afraid that that is not sufficient to allow you to control the ocntact.
The only area where you have any leeway is with the dog.
You are entitled to ask that he provide evidence that the dog is NOT the one which attacked someone and that in any event the chidlren are safe when the dog is around.
You shoudl try and discuss all these issues with your ex using Family mediation
I appreciate that this is not the news that you were hoping for, but it is the position that you face and I have to be honest
I hope that this is of assistance - please ask if you need further details