Resolved questions
Bi erkek cocuk ve babasi - bir erkek cocuk ve bir babasini

Bi (false word, true word "bir")
Bir erkek çocuk ve bir babası ( a boy child and his a dad)
it is mean 2 person..
No difference

"Bir erkek çocuk ve babası" : This description refers to two people who are a child and a father who is obviously known as the father of that child. In fact, it has a hidden pronoun, "onun" which indicates the "bir erkek çocuk". In other words, "Bir erkek çocuk ve (onun) babası." you should see the hidden pronoun "onun" at the end of the word "baba+sı"
For example:
"Sen ve baban." you should see the hidden pronoun "senin" which is represented by suffix "n" at the end of "baban"
"Sen ve senin baban."
"Ben ve babam." ---> "Ben ve (benim) babam."

"Bir erkek çocuk ve bir baba." This description also refers to two people who are a child and a father. But you are not sure if he is that child's father or not...