What are the chances that a father who never married the mother and also has two sons from a previous marriage could get shared physical custody of an 8-month-old child?

Truthfully, not too good. A baby is most likely to be left with the mother. Although if there are drugs or domestic violence involved it changes it drastically. Your two sons might be an influence as the court wouldn't want to separate the children and that is a positive for you. Also any property you might have owned before the marriage might influence the courts decision as if it is your house (purchased before the marriage) you all currently live in it is also to your favor. Good Luck.