1. They will give small events when hanging out/spending time with someone who you have a child with, for example when you take Kana to the festival and you have a child with her. there will be an option to take your child along. You can also interact with them in the daycare. Having a child won't affect any story too.
2. If you turn off Pregnancy, the child won't go "Poof". If you regret having a child, you can send them to the Artifact tho, in this case you can have another child with the certain mother.