Whilst the child is in the care of the non-resident parent and if they have Parental Responsibility, they will be able to make reasonable decisions about what that child can and cannot do. For example, they can decide what they can eat, wear and where to take them.
Provided there are no child protection issues involved there are no guidelines concerning how contact time can be spent and instead this relies on communication between the parents and agreed ground rules. For example, if your ex took your child to have their ears pierced or for a radical hair cut you might be upset by this. In this case, it would be worth sitting down together and talking it though to avoid further incidents of this kind.
When it comes to collecting children from school, it is necessary for the school to be aware of anyone who is allowed to collect your child and they will need to know who to expect in order to ensure their safe collection. For this reason, you will have to work this out in advance and let them know since they may be reluctant to release the child if prior arrangement has not been made.