Child support is determined by a legal formula which utilizes the parties' income and time share. There are many circumstances where support is due to the parent with less custody time. The court's interest is in some equalization of the standard of living between the two households.
Generally, claims for back support without an order or stipulation are difficult to sustain. However you mentioned that you had "come to an agreement". A court might order some payment depending upon the circumstances and legal arguments.
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1. You would be awarded child support.
2. Mother would still be awarded to pay child support unless she has some proof, like a medical disability showing she cannot work and earn and income. Child support can be based on minimum wage for your state.
3. Generally, there is no back support if there was no pending case or court order.
Until the incomes of the parties, the time, and other statutory factors are plugged into the formula that is used to calculate child support, no one can determine the amount of child support and who pays. An agreement without a court order is likely unenforceable. In any case the custodial parent and the time share has changed on 11-16-11.