The custodial parent is constantly trying to change the visitation order on her own to suit her ever-changing schedule. Can she do this?
No, only a judge can modify a parenting plan or residential schedule.
As time passes, keep in mind that some changes in the schedule are inevitable. Ideally, both parents should take part in any decisions relating to the residential schedule. Constant changes and/or significant changes based solely on the custodial parent's wishes, however, are neither legal nor reasonable.
Frequent schedule variations are also not fair to the child, who must constantly try to adapt to the latest schedule.
If frequent 'hijacking' of the residential schedule is having an adverse effect on the child or your relationship with the child, retain an attorney and file a motion for contempt, citing interference with the residential schedule.
My ex-wife is the non-custodial parent and has moved several hours away. Am I "in contempt" by not allowing the kids to call her long-distance? Shouldn't she be responsible for the calls?
You may or may not be in contempt, but regardless, you should support their relationship, even if includes some long-distance expenditures on your part. Generally, if something isn't in the parenting plan, it doesn't apply.
Also, by moving, some additional responsibility for communication is definitely borne by her. As the custodial parent, however, you have an obligation to promote contact between the children and the non-custodial parent.
If you were still married, would you call so the children could speak to her? Probably so, and although she may no longer be your wife, she is still a parent to the children.