There is no simple answer to your question. If there is a Will, the Will should name who the Executor of the Estate will be. That is the person who will handle the affairs of the Estate and submit the Will for probate. Just being named in the Will to receive an inheritance does not give you the legal right to be appointed as Executor. However, if the Will does not name an Executor or if that person declines the role, you may be able to get appointed as Executor. If your step-mother died...
If there was no will, the husband can petition the Surrogate's Court to become the Administrator of the Estate. Once appointed, any Estate property will pass through intestacy. In New York, intestacy succession is as follows: $50,000 and one-half of the residuary estate to the surviving spouse with the balance split equally among the four children.
The only way you can successfully appeal a guilty plea is to establish that the plea was not entered knowingly, freely and voluntarily. This means proving that the plea was somehow coerced or you were under some type of disability that you did not understand what was going on. Unless there was a trial, appeals from traffic court are quite unusual. Please note that your time in which to file a Notice of Appeal is 30 days from the conviction.
Unless the order specified a time period, there is no "typical" time frame. Since there was a garnishment order, it presumes there was some type of judgment as well. Normally, a judgment execution goes through the County Sheriff's office where the execution is to be made or the City Marshall's office if in New York City.