[MC5C]

Just make sure you are measuring from the top of the fret to the bottom of the string in each case. A digital (or other) caliper is a hard tool to use for that, it's just awkward if you are using the outside jaws, and highly subjective if you are using the depth measurement. Most people just use a machinist's ruler, since high accuracy is not really required.

The purpose of relief is to give the string vibration somewhere to go so the string can vibrate without hitting frets up the neck. People who play lightly don't need it, people who hit the string hard need a lot. It has very little to do with action at the nut. I play lightly and never strum hard, so I set my relief to close to zero, I don't even measure, I just look for a very little amount.

You can certainly measure the string height over the first fret pretty accurately with feeler gauges, but again extreme accuracy is not required. What I do is fret at the third fret, and look to see if the string is hitting the first fret, or if there is a lot of space. I want there to be a tiny amount of space over the first fret, just enough to say there is some, and just a tad higher for the lower strings. You can pick the string between the nut and the second fret (while fretting, maybe with a capo, at the third fret) and listen to see if it is hitting the first fret, or if you get a higher note when you fret at the second fret. If you do this, you don't need to measure but you'll end up with around the same as Dan recommends.

I set all my guitars to 12th fret action of 1/16" high E and 3/32" low E. I measure by eye against a machinist's ruler. I just set the ruler on the fret and look sideways at the string, and look for the bottom of the string to be in line with the line engraved on the ruler. You can buy a fancy ruler from Stew-Mac that does the same thing, but fancy.