Which organizations and individuals are invited for a review before the first flight of a new aircraft?
The level of contribution is your potential contribution to the particular flight test.

There are several levels:


Who gets a place in the aircraft? This should be test pilot(s) and maybe an engineer or two if the aircraft has enough space to carry them.
Who gets a place in the control room? This is the flight test engineer and systems engineers (experts for the engine, the control system, the landing gear and so on), also engineers familiar with the aerodynamics and flight mechanics of the plane. Generally, the number of people in the room should be kept small.
Who gets a place at the ceremony and press conference afterwards? Now only the test pilot is invited, and maybe the flight test engineer and the development project leader, but otherwise you will find mostly board members and PR people at the press conference. The engineers will be busy clearing the buffet at this time, anyway.


The responsibility to select who will get into the control room rests primarily with the flight test engineer and the test pilot. However, the whole aircraft development will already be observed by the FAA. From the FAA and Industry Guide to Aircraft Certification about the implementation of the planned flight test:


  During this Phase the Applicant and FAA work closely in managing, refining, and achieving their agreed PSCP to ensure that all agreed upon product specific certification requirements are met.


PSCP = Project Specific Certification Plan 

Normally, FAA engineers are reviewing the planning, but do not get into the details such as who exactly will participate in the first flight. The company developing the aircraft prepares a plan and FAA will approve it without much hassle if the company has a record of conducting flight tests. If the company is a startup, however, closer scrutiny must be expected.