Can someone have been on Cobra continuation coverage for the maximum period of eligibility (18months), then become insured with a new employer, retire, and be eligible for a brand new 18 month period of Cobra benefit coverage under the new employer's plan?

You should be eligible - you're on a different health plan than you were before, with a different employer . Ask your HR department to make sure - they are required to let you know if you are eligible.

Yes, Cobra is only a law that requires private insurance companies to provide various coverages,