You take both passports and show them to the Immigration on return in Thailand. You get stamped into your new passport. To transfer the stamps you need to visit your local immigration office after return. The Thai Embassy does not provide this service.
Talk to the hospital staff and get a letter that she is unable to travel based on medical reasons, with that letter and her passport you can apply in her name for the Extension of Stay, with a maximum of 90 days at the time.
Yes and that is based on working in Thailand which you are not allowed to do for a week as your Work Permit is expired, you need to organize quickly a new work permit or cancel this stay based on working.
That is the same, you can apply for the Extension of Stay based on unable to travel but have a plan B ready to go abroad, apply for a new Non Immigrant O visa and later the Extension of Stay.
If you are on an Extension of Stay, be aware that now the Work Permit is expired you need to cancel this Extension of Stay because you are not longer working in Thailand.