Almost all of your questions are answered in Pierre Gabriel's dissertation "Des catégories abéliennes".
He shows in a more general case, that the left exact functors between nice abelian categories are abelian und constructs indeed an exact "sheafification"-functor $T$, which is the left-adjoint of the inclusion functor. Have a look at Proposition 4 on page 348.
After that, Proposition 5 on page 374 implies, that $Func(\mathcal{A},\mathcal{Ab})/ker(T)\cong Lex(\mathcal{A},\mathcal{Ab})$, what you already mentioned.
I think $ker(T)$ are the so called "weakly effaceable" functors, but I am neither completely sure nor do I know any reference for that. Maybe someone can help you with this point.