Finally tested the fuel pressure and these were the results:
Note: the marks on the gauge are 0, 10, 20, 30, 40, etc... PSI. If I understand the fuel system correctly, shouldn't the pressure stay at 43.5 psi? There was no fuel leaking from the lines going to the gauge, so does that mean its leaking out of the injectors or the FPR? It seems like the FPR is working at least partially correctly, because it holds pressure at 43.5 PSI. I think the lack of pressure explains the hard start but not the stalling. You can also see that when the fuel pump starts sounding weird, it doesn't generate as much pressure.
Next up is visual inspection of the fuel pump assembly (and maybe I'll actually start the car with the fuel pressure gauge hooked up).