This depends if there are any moving restrictions included in your custody order or settlement agreement. Sometimes there will be a radius restriction which limits moving the residence of the child outside of a specific radius (i.e. "50 miles") or region (i.e. "tri-state" region). Even if this is the case you can always seek to have this restriction modified by the family court if there is a compelling reason such as work, health etc.
You certainly can represent yourself although you may want to check with your local Bar Assoc. to see if you are eligible for legal aid. In the interim visit the family court and petition the court for child support. There is no reason your spouse should not be paying child support.
You can certainly attempt to file a divorce without an attorney but chances are you will find yourself overwhelmed trying to navigate through the process. A basic divorce can be complex to those who are unfamiliar - the case you present would require many additional steps, You will need to make an application for substituted process service (likely publication) and that will require establishing due diligence in attempting to locate your spouse along with a formal application or motion. Once...
Distributions pursuant to a divorce are generally not taxable events. That is, there are no immediate tax consequences or penalties as long as the distribution was made pursuant to a Qualified Domestic Relations Order.
Legal separation is no longer necessary in order to obtain a divorce in New York. While in the past it was necessary in cases where grounds such as abandonment, cruelty or adultery were not applicable - with the introduction of the "no-fault" grounds, separation is not required. If you have been married for greater than 6 months you are eligible to commence a divorce action with no waiting period pursuant to DRL Sec. 170(7), irretrievable breakdown of the marriage.
If in fact you you were not properly noticed or served by the other party (or their counsel) you may move to vacate the judgment of divorce. You may have unknowingly signed an affidavit waiving any waiting periods prior to entry of the judgement. Seek attorney representation to resolve this issue.
While it is impossible for me to advise you regarding your entitlement to medical benefits without knowledge of the terms provided in the Separation Agreement, the agreement does not need to be filed in order to be binding . It would be a good idea ,to file it so that there is in fact, a record of the agreement.
If you do have a verifiable address for her you will likely need to have her served overseas (there are some unique rules with regard to foreign process service). If you cannot locate her, and you can document your efforts) you may be able to request alternate/substituted service such as publication or service on a family member. Please consult with an attorney before taking any action.