The law guardian's (now called attorney for the children) job is to advocate for the wishes of the children unless he or she finds that the children are not competent enough to have a position (too young) or the children's stated wishes would place them an imminent risk of harm. Just like you and your ex have attorneys to advocate for your position, the children have an attorney to advocate for their position as well.
If the father signed an acknowledgment of paternity or you were married at the time, you can simply file for child support. If not, you should file a paternity proceeding to have him adjudged the father and request child support. If there is no prior custody order, you have in effect sole custody so there is no need to file unless he tries to enforce parenting rights. Before doing anything, it would be wise to consult with a local attorney to go over your options.
He wouldnt which means one of two things: 1) the fact that his client will lose is not as sure as you think or 2) he doesnt have control over his client (meaning the client is going against his advice to settle). Settlement is the clients decision, not the attorney.
Sounds you may have a copy of a proposed Order. If you placed your agreement on the record in court then you don't have to sign an agreement. But these questions are better answered by your attorney who is familiar with your case.
Jurisdiction is state wide. If your daughter moved out of state, NY would still have jurisdiction until your grandson has lived there for six months. So you should file now. As far as venue goes, generally speaking venue is more proper in the county where the child resides, as that county is more likely to have the witnesses. However, if most of the witnesses live in the county where you live, you could make a cogent argument that venue is more proper where you live. Either way, you should...
A good idea is to try mediation but each have an attorney to advise you before each session. For instance, session number one with the mediators is about custody. Before your session, meet with your attorney to advise on that issue and so on and so forth. Much cheaper than litigation and you have the benefit of counsel to guide you. I often recommend it to clients. However, I would have one of your attorneys draft up the documents and the other attorney review them.
You will still need to pay the filing fees. You could go down to the county clerk and ask for the uncontested divorce packet, however I recommend consulting with a divorce attorney first since as counsel noted, you are the presumptive father and you need to make sure that she cannot come after you for child support.
An inquest is basically a mini hearing (usually one-sided unless your ex shows up this time) to determine the issues of equitable distribution, child support, spousal support, attorney's fees etc. So yes you will be granted a divorce but these issues need to be determine before your divorce can become final. But as counsel noted, these questions are best addressed with your attorney.