What is the optimal forecast for the following moving average model?
Your equation $y_{T+1} = E(y_{T+1}|I_T)$ clearly does not make sense.

In any case,
$$
E(y_{T+1}|I_T)
$$ is the best one step ahead forecast conditional on $Y_1, \cdots, Y_T$, whereas you're asked to find the best linear one step ahead forecast given $Y_T$. They are in general not the same.
The best linear forecast for $Y_{T+1}$ given $Y_T$ is
$$
\mu + \frac{\gamma(1)}{\gamma(0)} (Y_T - \mu),
$$
where $\mu$ is the unconditional mean of the series and $\gamma$ is the autocovariance function. In your case, $\mu = 0$ and $\gamma$ is that of a MA(1) series ($\frac{\gamma(1)}{\gamma(0)} = \frac{\theta}{1+\theta^2}, \; \theta = \frac12$). 

In general, the best linear forecast for covariance-stationary time series can be obtained by solving the Yule-Walker equations.

For best linear forecast, the assumption that innovations are i.i.d. is extraneous.