There is no 'family application'. Each person needs to file a separate application and pay the fee. It is best to have your wife file in India and to file out the application for the children. They will worry if someone applies for a child in a different country and it introduces an undesirable layer of complexity to the application.
Proceed as follows...
You, as the primary, fill out the application and use the remarks section to explain that you are primary and will be taking care of the child during the visit. Send your GWF number to your wife.
She fills out the application and explains that you are the primary. She should put your GWF number in the remarks section. Repeat the procedure for the child.
When you print out the application, use a pen to write your wife's GWF number and your child's GWF number. The dependants should do the same.
The normal rule is that only the primary should submit evidence. However, your application will be decided in New York and the family members will go to the visa section in India. Make a copy of your evidence and send it to your wife. On the top of each page write "COPY" and your GWF number. The two visa sections will email each other and sync everything up in Proviso so that the two ECO's can see the bigger picture.
IMPORTANT: The VFS personnel in India have been removing evidence from people's bundles without their knowledge and this has caused refusals. So it's vital that your wife include a transmittal letter that itemises your evidence. Send a copy of the transmittal letter to their contact email also.
Because it's a family application submitted to two visa sections, the application will be more complex so their standard turn-around times will not apply.
For your question about the different centres, at the end of the application the programme will tell each application which VFS to use for their biometrics and submission.
Member 'pnuts' (to whom thanks) has pointed out that they do not want to see flight reservations or flight tickets or hotel bookings and the like.
The visitor rules are in Appendix V of the Immigration Rules.