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Each parent's responsibility for the child's expenses is in the child support order entered by the court - usually coincides with the property settlement agreement. Whatever the order says is what each parent, custodial and non-custodial, must do financially and for insurance coverage.
If you're the non-custodial parent, and you've been ordered to pay a certain amount of child support, and you do so, then you have no other legal obligations. Also, whichever parent was ordered to maintain insurance for the child must do so - if the expense is 50/50, each parent must comply.
Take a look at the child support part of the order and you should find your answers.
You can file a petition in the original court asking to modify the child support obligations to include that the father must provide insurance. It doesn't matter that the father has moved. It also doesn't matter that the father hasn't had anything to do with the child - he's still obligated to pay.
As for the $1500 credit, I cannot tell you what's going on there. The child support division of the Dept. of Human Services, whom I assume collects and forwards payments from him to you, should be able to give you an itemized accounting.
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