The person to whom a blue badge is issued must be either the driver of the vehicle when it's used or a passenger in that vehicle. So the answer to your question is 'No.
No. It is awarded to your sister only as a driver or passenger. When my father went into a Macmillan hospice a few years ago. Immediate family was issued with parking permits for the grounds. Ask at the reception to see if they provide a similar service where you are.
No, PP. You can use the car that is taxed as disabled to do their shopping and run other errands on their behalf (so long as relevant insurance is in place) but you can't use the blue badge.