Sparky,
OK, I would suspect one issue is neutrals from multiple (at least more than one) circuits connected together downstream. AFCI (GFCI) doesn't care with no load. When the light is turned on part of the neutral current heads back through another path and the device trips. MAY also be a cross between neutral and equipment ground. (Found a receptacle recently with the neutral on the yoke and the EG on the neutral terminal...pretty bad!) On the offending circuit, remove the neutral from the bus and see if you can read to the bus from the free end. Shouldn't be...
The 14 ohms is interesting but I would lean towards a transformer device. What does the circuit read if you flip the leads around on the meter (change the polarity)?
I have pesonally received a hefty belt from a trailer door while standing on the wet ground....keep digging!!
