Is a daughter, who leaves her elderly mother—for whom she is responsible—with one of her relatives for a week to go on a trip with her family, while covering the mother’s expenses during this period, considered sinful?

There is no harm in a daughter traveling for a short period and leaving her mother with someone who will care for and serve her until her return, especially if this is done with the mother's consultation and consent. The daughter must ensure her mother's well-being and confirm that she is being cared for during her absence. The daughter should also thank Allah for enabling her to be dutiful to her mother and ask Allah to help her in this regard.