Thank uou for all your advice, I have found the way to prove proposition 2.

Just to check that, in a balanced category, if $f \colon A\rightarrow B$ induces a surjection(in the sense of set theory) from $Hom(C,A)$ to $Hom(C,B)$ for any object $C$, then $f$ is surjective. I believe that it will give me the desired result.(Hope that I am on the right trend)

Thank you for your reference, for proposition 1, I proved it successfully, but I was now stuck at proposition 2, I nearly have no idea on how to relate the definition of extremal generators with generators via the definition of balanced category

Thank you for your answer, I would like to know that is there any further reference and proof for the statement?