It is possible to request that refund, however DMV will issue it to the name of the estate, so you would have to work that out in a way for you to receive it then.  To request the refund of your recently passed mother's registration fee, you must surrender the vehicle plates at a DMV office and request a transfer receipt, not a refund receipt.  You need to have the executor or executrix of the estate send a completed and signed Request for Refund [1] MV-215 with "deceased" on the form, a photocopy of the death certificate, and the transfer receipt to the address listed on the Request for Refund form.
I need a little more information about New York State licensed TNC companies.  Is there a minimum age requirement?
Yes, of course. You need to be 19 years old or older, to be a tnc driver.
And Can I use any vehicle I want?