I find it very odd that the school called dad and said come and get your children because they've had an argument with Mum.. and there is no more to it than that.
What if the parents were together? They wouldn't call dad and say "I think you should do pick up today your DS had a row with his mum this morning"
Sounds to me like your DD said more about the argument than what is written here. And, if so, there will be a report somewhere to document this.
Also, it seems from your posts that the school weren't aware of the problems you have had with their Dad because you didn't want to be seen to be slagging him off. What are your concerns about his parenting?
To be honest, I feel the blame lies at his door for not telling you he had the children, and also a bit at your DD's door as at 10 she should realise not to exaggerate arguments or go home with Dad when you are expecting her home. My DD would get a bollocking for this. Not because she;d gone home with Dad, but because she deviated from the plan that she would walk home and as a result I was incredibly worried about her. It's not very responsible behaviour.
In answer to your question, at our school we have a form to fill in when the child registers there to ask who the main carers are and if there are any other people who would regularly collect the children. If someone is listed on there, no one asks if they have PR, they just hand them over.
For example, I am on there and DDs dad is on there (They have no idea if he has PR or not) and his wife and my husband are also on there. Neither has PR but if they turn up to collect DD no questions are asked.