3 parent households are not that common. I think what is more common is if there is two parents and a live-in relative, like an aunt or uncle or grandparent.
Surrogates don't always have access to the child. Some surrogates might not even be the child's biological mother. Straight couples also seek out surrogates and surrogates can also come from egg donors.
Depends on if it is an open or closed adoption. If it is a closed adoption, legally, the birth parent can not do this. In cases like that, it has to be the child being an legal adult choosing to seek out their birth parent. In an open adoption, it's rare that the parent would move in. Tho some will still met with their kids from time to time or get updates. My adopted siblings are from another country and my parents stayed in contact with their birth mother and siblings. But I have also met people who, usually from in country adoptions, who disliked meeting their birth parents due to is being common that kids are given up due to the parent having a mental illness or issues with drugs.
I think a story about three parents could be interesting as long as it isn't some sister-wives thing. Some people see Steven Universe as Steven having three moms, so it can work.