The Gym Membership seems crazy, UNLESS the child has some type of medical condition that is being addressed by the gym membership. i.e. if the child is obese and the gym membership is part of an overall medically supervised weight loss program.
I'm not saying it would be a sure thing you'd have to pay in that case. But I think a good argument could be made that, in that case, the gym membership would be a medical expense and should be treated as such.
As for private school, unless the order addresses it, you don't have to pay for it. However, the CP could take it to court and get private school included. If the child was in private school before the divorce, there is a reasonable chance that you will be paying at least part of the tuition. While private school is not something the child NEEDS, the courts want to minimize the changes that the child has to go through after the divorce. So, if you guys made the decision to put the kid in public school during the marriage, and they are used to going to that school, it is very likely that the courts will want to continue that arrangement.
To sum up; you don't have to pay for anything unless it is in the order. However, the CP can always take you back to court and have things added to the order if the court decides it is reasonable.