No problem. TNC stands for transportation network company, a term defined in a recently enacted part of the State Vehicle & Traffic Law. In other words, a TNC is a "rideshare company" licensed by DMV to use a digital network to connect passengers to TNC drivers for prearranged trips.
To do that, I guess i need permission from you, isn't?
No. DMV does not have nothing to do with your clearence for being a TNC, but it could be something required by a TNC Company.
Is there a phone number I can call to request a closest family member form?
You can call the DMV Revenue Accounting Unit for that form at 518-474-0902.
What is the exact purpose of that Next of Kin form?
The Next of Kin form is used if the estate of the registrant is settled.  In that case, the refund is issued in the name of the next of kin.
What should I do if I decide to transfer ownership of my recently deceased mother's car?
You can use PDF [2} MV-843 as the checklist to help you transfer the ownership of the vehicle.
Since my mother has passed, who can issue a refund to her estate for her registration?