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Yes, she can ask for it. She can also ask for step children. If your employer does not allow it for non-biological children then you can voice that to the judge. As for covering her, it is up to you to agree or the judge to agree. If you agree to that, you should seek some sort of concession on any spousal support or other issue being negotiated.
If your company insurance does not allow it for some reason, the signed divorce decree serves as a court order that they would have to comply with. Alternatively, you can pay her the monthly cost of her own insurance premium but I imagine that would be more expensive.
In a divorce case, technically everything is on the table for negotiation. You just have to either negotiate an agreed divorce or let the judge rule on these open issues. Let me know of any specific questions you have.
I saw that you posted additional questions. If you like, we can do a phone call for an additional charge so that you can get all of your questions answered.