I think you have something like 30 days to apply for Medicare and Medi-gap policies without medical underwriting after you become eligible for Medicare. If you delay, you can get much higher premiums because of the medical underwriting and there are some additional fees in Medicare for getting it after you are first eligible.

You are first eligible two years after you apply or two years after the decision you are disabled. Usually the first criteria is used but if the administrative person who took your application didn't put down that you had a protected filing date I think sometimes the latter applies.

CORBA benefits are only for a limited period of time so generally it's better to get Medicare while you can. You can try and negotiate with your employer that COBRA is the primary and Medicare the secondary which get you better benefits but that depends on your employer. I do know one person who got CORBA long term but that's an exception to the usual rule.

I wrote a response to you in the TOS forum about what Medicare covers for TOS.