Let $M$ be a W*-algebra with unit $1$.
Q1) Does there exist a normal positive linear functional $\phi$ on $M$ whose support is 1?
Q2) We know that there is a unique central projection $z$ in $M^{**}$ by which every bounded linear functional is decomposed to a normal and singular part: $$M^*= \underbrace{zM^*}_{=M_*}\oplus^{\ell^1}(1-z)M^*$$ True or false: $1\le z$ ?!
As for your first question let $M=\ell_\infty(\Gamma)$ where $\Gamma$ is uncountable. Certainly, $M_*=\ell_1(\Gamma)$ and each element in $M_*$ is countably supported.
I am not quite sure what is the second question about. You want a projection dominating the unit?