Is it permissible for a mother to transfer ownership of her house to her youngest son as a "trust" to prevent future problems among siblings and to ensure the house remains a refuge for everyone, given that this son is the family's provider and is unmarried?

Your mother is not permitted to register the house in your name as a trust (amanah), as this could lead to dispute. It is better for your mother to appoint one or two righteous men to carry out what is necessary according to Islamic law after her death, and for her to have witnesses to the will and to officially document it, in order to ensure that the house passes to its rightful heir according to Islamic law.