I believe that this entire process is controlled by 18 NYCRR 347.9. There is no way to directly link to this regulation, but you can reach it through this link.
The regulation does not discuss what happens if the employer fails to serve notice on the employee. However, the presumption is that if you are actually being garnished, then you must know that it's happened. So, if you haven't actually been garnished, or the garnishment occurred, within the notice period, so as to limit you notice of the garnishment, then I believe the court would have equitable authority to extend your time to file a contest over the income execution.
As for recourse, it's wrongful garnishment against the attorney (assuming that this is in fact a wrongful garnishment). If the court were to refuse you the opportunity to contest the garnishment, then you could still ask the court for a determination of arrears, and obtain credit for any overpayments.
I think you may have difficulties holding the employer liable -- though it's not clear in any case law. Ordinarily, negligence does not lie for pure economic injury -- which is what is in play here. However, you may be able to argue that the employer has a special duty, because of its obligations under NY law to not make any unlawful deductions from employee pay. This could be a "rathole," and your employer could make your life miserable, if you start filing legal actions against the employer. Hopefully you can resolve everything with the family court re opposing counsel's alleged errors.
Hope this helps.