What is the appropriate age for a child to sleep in a separate room, and is it permissible for the mother to sit with him in light sleepwear, and what is the psychological and moral impact of that on the child, and at what age should the mother observe modesty in front of her children?

There is no harm in a child sleeping with their parents at any age if the 'awrat (private parts) are secured. It is preferable for a child to sleep with their parents at this age due to their need for care. When the child reaches the age of discernment, it is permissible for them to sleep in another room if no harm results from it.