Degree of cure is usually measured by DSC (differential scanning calorimetry). The instrument measures the heat flow into or out of a very tiny sample (usually 10 to 50 milligrams) as the temperature is ramped up (usually 10C/minute). The total heat of the curing reaction is measured on uncured material. If any reaction heat
(exotherm, delta H, in joules per gram) can be measured on a an already cured sample then it was not fully cured. The percentage of cure can be calculated by dividing the residual delta H by the total delta H.
Degree of cure can have a very strong effect on strength and modulus. Some materials can be over-cured as well as under-cured. Over-curing causes excessive crosslinking which makes the polymer more brittle.
Degree of cure is not relevant to carbon-carbon because the polymer has been pyrolyzed into carbon and isn't a polymer anymore.