All you can really measure is the DC Resistance, but that tells you what the impedance is (multiply DC resistance by 1.25), which gives you a rough idea of the output of the pickup. The higher the impedance, the higher the output.
You can use resistance to compare similar pickups. The strength of the magnet and configuration of the coil(s) matter a lot too.
Resistance is a very good way to decide which pickup to use in the bridge position, but it's less usefull when deciding whether a Lace Sensor will be as loud as a PAF. It also says nothing about tone.
A G&L Field coil pickup uses stronger magnets and less resistance to arrive at it's output.
People make too much of coil resistance when selecting pickups.
All that said, it's still a useful piece of info to have.
You can measure the output voltage (Millivolts AC) but there are so many variables (how hard the note was picked, string gauge and age, etc) that I doubt that it would be very useful info. A typical out put voltage from a Bucker might be 150 to 300 mV AC.
Yes. The most common way a pickup fails is breakage of the wire that makes up the coil. If you put an ohmmeter across the terminals of a pickup which has failed in that way, you'll see an open circuit. A good pickup will read in the k Ohms range. A shorted pickup (also a failure) will read a very low resistance.