You will not accurately measure the voltage across the LED because it is pulsing at a high frequency. Your meter will measure the average voltage which might be half or less of the actual voltage.
It depends. On what kind of meter you use, and what the frequency is, for instance. What do you suggest to solve this?
Another point. If I'm inaccurately measuring the LED current but all I need is a comparison between two circuits, then assuming the inaccuracies are the same in the compared circuits, they shouldn't be a problem. The comparisons are what I'm after, not an accurate measurement. One is brighter and I measure higher LED current than the other circuit. I don't have a decent accurate light meter, so this seems to be an easy and cheap solution to making a comparison of actual light output.
I'm on the fence about how using the resistor in series with the LED works or if it's inaccurate. What I did was try measuring with a bypass capacitor of various sizes across the resistor, and it didn't seem to change the measurement. Then I decided that putting a 100 uF or even larger across a 1 ohm resistor was not enough, since the reactance of the capacitor was still more than 1 ohm, even at 100kHz. So I put a 100k and 0.1 uF or larger RC low pass filter between the resistor and the meter. Well, it didn't change the measurement either. But what led me to do this was that I was calculating (and other experimenters were too) an efficiency of 85 to 95 percent, and I thought that was out of line and it was due to some inaccuracy of the voltage measurement across the resistor.
Oh, one other point. The amount of LED output is not related just to the pulse height. If the pulse width varies, so does the LED light output. And another thing is that the LED light output is not linear in relation to the energy contained in the pulse. So if I get an accurate measurement it may not reflect the exact relationship to LED light output. Some complex issues, so I hope I am making myself clear?