When the inlet pressure falls close or equalizes with the outlet pressure what happens to the flow rate though the regulator?
At what differential (pressure inlet vs outlet) the valve will fully open?
It depends on the specific design of the regulator. In an ideal regulator, the flow and output pressure will stay constant as long as the inlet pressure is adequate. But most regulators have "droop" which is what you are asking about. "Instrument grade" regulators have pneumatic amplifiers, which consist of a very sensitive pilot valve that controls the position of the main valve.
If you study the linked data sheet you can learn a lot about regulators.