G.♣.
For now you have to share custody with your ex, but you can take him to court and ask for a change in custody. Until you are ready and able to do that, keep notes on the things that are harmful to the children. As hard as this is going to be, though, you have to separate the things that annoy or insult you from the things that harm them. Attacking your faith, telling the kids to call you "birth mom" and trying to convince your daughter she doesn't know which role she played in the Nutcracker are examples of him being petty and just trying to upset you. He's not being a very good parent, but he's not damaging them, either. Try to focus on the things that are truly harmful to the children, like denying them medical care.
Remember, when the kids are with him, that is his time. He doesn't have to tell you what they are doing, and he doesn't have to let them talk to you every day, even if it is to read the Bible or pray. You can do that with them on your time. He doesn't have to do it on his time. Even if you are only able to share your faith with them during your time, it will still have a huge impact on them.
Please stop wasting your time and money buying your ex things. You don't need to babysit or buy them gifts or send them things. When he told the judge you were alienating him, the judge took that to mean you weren't letting the kids see him. He gets to see the kids 50% of the time, so he really can't accuse you of alienating him.
The children deserve to have a relationship with their father. If you are truly concerned that he is abusing them (not the same things as saying bad things about you), talk to your lawyer about that. Otherwise, try to make this work. You don't have to like him. You just have to be supportive of the fact that he is their father.
ETA - You're right, I did not know what parental alienation was. I just looked it up, and it does not mean that you have to buy your ex things or give him gifts or send him a Valentines Day card or buy him tickets to six flags or invite his wife to lunch or any of the things you mentioned.
I do apologize if the court did order him to let you know whether or not he chooses to take the kids to the park for the afternoon. That does sound extreme! It would be nice if he told you what the kids did while they were with him, but I wouldn't expect a detailed itinerary.
My point wasn't that he's a great parent or everything that he does is ok. My point was, before you go to court again, make a list of the things that are noteworthy and that actually harm your children. Not giving you a detailed list of everything they do should not be on the list. Not allowing you to read a gospel passage to them over the phone during his week should not be on the list. And he can't say you are guilty of parental alienation because you only bought him 4 tickets to six flags when he really needed 6.
I'm just saying, be smart about this. You need to be able to convince the judge that having this much time with their father is a bad thing for them.
My husband and I disagree on politics. His way of saying why he doesn't like my candidate is to say things like, "He's Satan," or "I just don't trust him." I told him, you have to convince me that the candidate is wrong about an issue. Convince me! Well, you're going to have to convince a judge, and judges tend to want you to not waste their time on petty things. So make sure the concerns you bring forward are valid and convincing, that those concerns truly to have negative effects on your children.
Just make sure you are careful about the issues you choose to focus on.