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That depends on your STD program, which isn't controlled by state or federal law. It's controlled by contract law, so all that policy has to do is cover in accordance with it's policy terms.
That being said, most STD policy terms only pay out to those who are or would be active employees if they were not out on STD. So, if you are being laid off, that means that you are not or would not be an active employee even if you were not on STD during the month of September.
Therefore, with that understanding, no you would not be entitled to payments during the month of September, because you would no longer be an employee.
No, because it isn't health insurance. It's STD. Health insurance has such a vehicle, called COBRA, but STD does not. It just wasn't included in the COBRA law as something to be continued beyond employment.
The definition of beneficiary is entirely controlled by the contract, and if a laid off employee is not considered a beneficiary under their contract, that is unfortunate the end of the discussion.