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A parent can't be made to take custody of the child(ren).
I would recommend filing a custody action and thereby establishing a custodial order as to custody/visitation that both parents have to adhere to - if one parent continually violates the order or is unreasonable in carrying out the order then the other parent has the ability to obtain primary custody.
That's a really tough question - you want to be there for your child but you don't want to be on constant call - it's really a judgment call on your part. I would be ignorant of your situation of I told you otherwise. I can't answer that one for you.
Legally, if you are on call for her, keep a diary of all happenings just to protect yourself later and to remember dates and events. You don't set any legal precedents if you take him or not as per the agreement - it's just the relationship between it all.