You can refuse to give the custody of your child, let him approach court for this where you can challenge the same as per law.
If the mediation fails and the case is reverted to court, you can insist on orders towards interim maintenance, which the court will pass it on merits.
If the child custody case has not been filed by him and the child is in your custody only, then you need not worry about it, let him file the child custody case and the court will decide on it on merits.
There is no hard and fast rules that the child has to be with one of the parent only even after attaining 5 years of age, the court will decide based on the facts and merits on both the sides.
If you have lost confidence in your present lawyer, you can very well change the lawyer but no lawyer can give you assurance for a faster disposal of the case.
If the case is transferred to Delhi, the parties to the case have to attend the case in the court where the trial is going on.
Interim maintenance would be on monthly payment only and not a lump sum amount.
The maintenance amount would be paid by him to you before the court on the dates of hearing.
No agency working for women welfare can hasten the case or have the powers to pas any such order, and no agency is above court, hence since you have already approached court, you have no option than to follow the court process.