Welcome back to the Wefugees Community and thank you for your very interesting question.
First of all, you can apply for the permanent residence permit as soon you fulfil the requirements. You don't have to wait until the current residence permit is expiring.
Please be aware that only children from 16 years onwards can apply for a permanent residence permit. Younger children's residence is usually dependent on their parents' status.
If you living as a family unit, you can also "secure" your family's livelihood and cover the life expenses. You do not necessarily have to be married.
On your last question: The child's mother may apply for asylum for your child. If he/she is recognized as refugee, he/she can get the blue passport as well.
Do you agree dear @mbeon-fardeen?
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,