Existence of faithful normal state Does there always exist faithful normal state on center of von Neumann algebra? 
Further, in type II$_{1}$ von Neumann  algebras are tracial states coming from the center valued trace?

The first question is the same as asking if every abelian von Neumann algebra has a faithful normal state. The answer is easily no, by taking for instance the example in this answer. 
As for all tracial states coming from the center-valued trace, it is not entirely clear to me what you mean. If you have 
$$
M=\bigoplus_n A_n\otimes N_n,
$$
where $A_n$ are abelian and $N_n$ are II$_1$-factors, the center-valued trace is induced by 
$$
\Phi(\bigoplus_n a_n\otimes x_n)=\bigoplus_n a_n\otimes 1. 
$$
Tracial states are obtained by taking $f_n\in S(A_n)$ and $\tau_n$ the unique trace on $N_n$, and letting
$$
\phi(\bigoplus_n a_n\otimes x_n)=\sum_n f_n(a_n)\tau(x_n).
$$
I cannot immediately say in what sense $\phi$ "comes" from $\Phi$. 
