Regulator is a diaphragm that is connected to the fuel rail that either allows fuel to return to the tank or block the return flow in order to regulate fuel pressure.
First thing I do with any suspected fuel problems is check fuel pressure, with key on, cranking but not starting, and then running idle and revving. It's quick and simple, provides much info, and a fuel pressure gauge is cheap, should be able to get one for a few bucks or rent one. It attaches to the fuel rail on a fitting that looks like a valve stem on a tire. This will show problems with injectors, pump, filter, regulator, etc. Refer to maintenance manual for the factory fuel pressure requirements for your car in various modes of operation.
Fuel filter is also a quick easy fix and is regular often overlooked maintenance anyway.
Low pressure could mean an obstruction, pump/regulator problem, or even faulty injectors or injector driver module (would have to check this on a scope) Having to hold the throttle open to start the car, then having it stumble on load *sounds* like its running too rich on cold start.
Not familiar with your car to know the extent of on board computer diagnostics, but a scan tool to check the problem logs on a EFI car is also invaluable.