First and foremost, everyone has to be on the same page. Is he even fully committed to being in a relationship with her again? Secondly, find out what he wants from you, her, and you guys as a couple. Either by talking directly to him or conveying it through her. I was the hinge in a V before I knew what it was called. My girlfriend had no interest in a relationship with my DH, and it worked for a decade and some change. Now, it's a triad.
As far as children, ours are 4 and 7 months. I gave birth to them, but they view our girlfriend as a second mother. The 4 year old even calls her mum. Our children are young, but I haven't noticed any negative effects. Personally, our children have an overabundance of love from three parents. There's never an issue of mummy and daddy are working so they'll miss my ballet recital. At least one of us is always present. I'm sure some of the other members have older children and can tell you more about their experiences.
Electing to keep it quiet or telling people is a personal choice and depends on comfort. In my case, I decided to tell because I always felt like she shouldn't have to be my dirty little secret. We love her, and she's part of us and our family. I never wanted her to feel left out when my DH and I received wedding invitations or any invites. Instead of +1, it is +2. Do what feels comfortable and makes you feel right. In a triad or more, it's not uncommon for some of the parties parents and family members to not know. They might think he/she is just a best friend. If you know your friends and family, think about who you can trust if you can keep it a secret.