When the USPTO enters an insignia in the database, does it first make an official determination that the insignia is the official insignia of the Native American tribe that made the submission?

No. When it receives a request to enter an insignia into the database, the USPTO does not investigate whether the insignia is actually the official insignia of the tribe that made the submission. Instead, the USPTO complies with any request to enter an insignia into the database, provided that the request is made by a federally or state recognized Native American tribe, and provided that the request includes each of the necessary elements for it to be entered into the database.  See listing of elements necessary for the insignia to be entered into the database in the below FAQ.