My mother just went under hospice care at her home. If she needs to go into a facility full time, do all of her assets go to that facility?

All of your mother's assets will not automatically go to any facility that your mother goes into.  However, you will be expected to use those assets every month to pay for the care that she is receiving.  Some of the care will be covered by Medicare through Hospice, but you will be responsible for providing for room and board.  You will need to liquidate investments as each monthly bill for the facility becomes due.