If he has temporary custody, I surmise that there is a pending court case. Nothing is cast in stone yet. You need an experienced family court attorney to represent you. It sounds like you are now "paying" for having removed the child from the jurisdiction and from the father. Your position needs to be explained to the court and you need to present strong arguments as to why it is in the child's best interest to be in your care. A strong parenting plan, including appropriate parenting time...
The other attorneys are correct. You should be getting more. Further, you should get a support order from the court. That protects you if he stops paying. Even if you are willing to accept less that the statutory amount, you should do it through the court.
There are no formal guidelines, other than that the supervisor is to be present at all times during the visit. It is usually best if the supervisor maintains a bit of distance and allows the children to interact freely with the parent. Generally, there is no harm in the father bringing the children's grandfather to one, or some, of the visits. But the time is really meant for him to enjoy with the children. As for the location, the supervisor has no obligation to supervise. They are doing...
File a police report and let them investigate. If they bring the case to the DA for prosecution, they will have access to his arrest record. There are limits, however, on the introduction of prior bad acts in a criminal trial. Nonetheless, let the professionals handle it. Just start by going to the local precinct and filing a report.
It's never a good idea to get involved with a married man. If he were going to leave his wife, he would have done it before getting into the next relationship. At this point, you have little to gain from this relationship. Take the advise of the previous attorneys and move on.
Your sister should come to NY and file a Guardianship petition, naming both you and the other parent as respondents. If both parents agree, the court will take her fingerprints and run a background check. A lawyer will be assigned to the child. If he is over 14, the court will require that he sign a consent called an Order of Preference. Assuming all is clear with the background check, and the other parent also consents, a guardianship order can be issued on the next court date.
You don't really provide enough information for an attorney to give a complete answer. Is this a child support matter? If you are just providing requested information, the court may accept the printouts. If you are conducting a trial, they would not be admissible. You would need to subpoena certified documents from the bank.
Unless you were directed by the court to mail it in, I would fill out the form completely and bring it on the next court date and present it directly to the judge. These things always seem not to end up in the case file when they are mailed in. Make a couple of copies. Keep one and provide another to opposing counsel if you are directed to.