Regardless of year 27 ohms is a little low for either, 30-200 is the absolute maximum range for the 83, but at 27 your well below the average which i normally see around 150-170 ohms on them. I can go grab one off the spares shelf later and measure it if you wanted a comparison.
Question 2... The pulse generator, normally what i call the pickup coil, is just that a coil. So the higher resistance the better on these, the lower the resistance means generally there is shorting between turns of the coil and its starting to break down on its insulation. A coil generally goes bad by either shorting and losing its resistance becoming a dead short, or it opens up and has no continuity.
The advancer pickup coil is the same as your stator/source coil. A magnet, in this case on the advancer, passes the coil creating a pulse of voltage that the CDI sees as a trigger for the ignition. So if the coil has a low resistance and can't build that voltage pulse then the CDI won't get a trigger to fire. The pickup coil is so cheap i generally just replace it if i suspect it, or at least swap it with a known working one to rule it out. However you can test it with a test light or an analog voltmeter set to AC, the digital meters won't respond fast enough to the pulses unless you build a peak reading adapter for it. With the test light you will see flashes as you crank connected to the output of the pickup coil, with an analog meter you will see the needle jumps.
I take it your having a problem such as spark/no spark issues? I would suspect the CDI before the pickup coil, very common problem.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....