I think the mention of "I have to be like a bill collector" was an expression of frustration about how hard it is to get her ex- to pay, as opposed to an expression of a desire to sell the debt to a collection agency.

If the court ordered payments to (retroactively) begin on June 4 then the arrearage would be the total amount of payments due since June 4 less the amount your ex- has paid. There may be interest on the arrearage; if so, probably only since the order was entered.