Let $(M; \tau)$ be the hyperfinite $II_{1}$-Factor and consider a W${}^{\ast}$-subalgebra, $N$.
Is there a (trace-preserving) conditional expectation, $E:M\to N$?
Considering, now, a more general context for $M$, given a conditional expectation, $E$, and a(n) (positive) element $a$ with $E(a)=0$, does it hold that $E(a^{n})=0$ for all $n$ (equivalent: $E\upharpoonright C^{\ast}(a)=0$)? Does it hold that $E\upharpoonright \{a\}^{\prime\prime}=0$?
For your first question: yes, every W*-subalgebra of the hyperfinite II$_1$-factor is the range of a trace-preserving conditional expectation.
For your second question. If $a\geq0$, you can use that $E$ is positive: as $a^2\leq \|a\|\, a$,$$ E(a^2)\leq\|a\|\, E(a)=0;\ \ \ E(a^{n+1})\leq\|a\|\, E(a^n). $$ If $E$ is normal, then it will be zero on $\{a\}''$.
For $a$ not positive, let $M=M_2 (\mathbb C) $, $E$ the projection onto the diagonal, and $$a=\begin{bmatrix} 0&1\\1&0\end{bmatrix}.$$ Then $E(a)=0$, $E(a^2)=1$.