Just take a blue power LED with a decently-sized die. That's basically the same as your white LED with the phosphor removed (again, it's not a UV LED hiding underneath the phosphor).
Which for your measurement is basically irrelevant. The net efficiency is probably significantly higher with the phosphor removed. Keep in mind that a LED used as a photodiode is sensitive to wavelengths shorter than its peak wavelength when used as an emitter. So the blue diode inside a white LED is sensitive to UV, and in your use case, all the phosphor does is convert a little UV to green and red light. This is essentially lost to the sensor diode, which is blind to those longer wavelengths.
So in short, you're better off with a simple blue LED. Of course, white LEDs are plentiful, so there's a practical reason why you might prefer those; that's fair enough.
Or just build a simple transimpedance amplifier. Plenty of circuits all over the net.
Not sure what you mean here. Many light sources will indeed flicker at e.g. 50/60Hz grid frequency, or at a substantially higher frequency depending on how they're driven.