You will not need to attend the compliance conference. In Nassau and Suffolk Counties, you need not make any further Court appearances. In Queens, you are required to appear for an inquest in order to obtain an uncontested divorce.
He is not necessarily entitled to half, but has an interest in the property due to the fact that you purchased it in joint names, and he makes contributions, albeit not direct contributions to the property, but indirect contributions by paying other family expenses. You could get a credit for the deposit which came from an inheritance if you can trace it with a clear paper trail. It may be tainted, however, if you put the inheritance money in a joint account and comingled it with marital funds...
It is very difficult to rescind an agreement based upon fraud. If you can get past the six year statute of limitations, you would have to prove that there were material misrepresentations of facts made at the time of entry into the agreement. You would also have to show that you availed yourself of all discovery vehicles or knowingly waived them at the time of the agreement.
The answer is no. Gifts are not subject to ed. However, if the gift is in "cash", keep in mind that the burden is on the party asserting the claim to prove separate property. Since "cash" leaves no paper trail, it can be difficult to prove its source.
There is a chance, particularly if the recommendation in the forensic report is not sound. More and more these days the forensic examiners do not make recommendations. There are many who feel that is only for the Court to decide after hearing all of the evidence.
If you filed for divorce, I assume you already have an attorney. If not, you should get one. If you do have an attorney, you should seek his or her counsel. If not, you should look for a new one. A lawyer cannot give you advice online in this forum. You need to schedule a consultation to obtain legal advice.