Is the questioner obligated to sit with her incapacitated mother, who resides in her brother's house, and what are her duties towards her, and will Allah be displeased with her if she does not do so?

Allah Almighty commanded kindness to parents and linked their right with His right, especially in old age. The questioner has done well in her dutifulness to her mother, and she should visit her from time to time. Her mother's stay in her elder son's house might be better due to the questioner's preoccupation with her husband and child, and what people commonly recognize as connection and dutifulness is sufficient for her maintaining ties.