I'm still not quite sure how part (a) is different from part (b) but, from your comment above it appears you are now only asking about part (b), so:
If $\psi[\hat{\theta};(X,Y)] = 0$, then
$$ \sum_{i =1}^n(Y_i - \hat{\theta} X_i) \> = 0 $$
So,
$$ \sum_{i=1}^{n} Y_i = \hat{\theta} \cdot \sum_{i=1}^{n} X_i $$
Therefore $\hat{\theta} = \overline{Y}/\overline{X}$, the ratio of the sample means, satisfies $\psi[\hat{\theta};(X,Y)] = 0$. Regarding unbiasedness, it is easy to see that
$$ E( \hat{\theta} ) = E\left( \frac{ \sum_{i=1}^{n} \theta X_i + \varepsilon_{i} }{ \sum_{i=1}^{n} X_i }\right) = \theta + E \left( \frac{ \sum_{i=1}^{n} \varepsilon_i }{ \sum_{i=1}^{n} X_i }\right), $$
therefore $ \hat{\theta} $ is unbiased. Since this is homework I will leave it to you to explain why that would make $\hat{\theta}$ unbiased and why the independence between the $\varepsilon$'s and $X$'s (and that fact that $E(\varepsilon_i)=0$) implies that $E \left( \frac{ \sum_{i=1}^{n} \varepsilon_i }{ \sum_{i=1}^{n} X_i }\right) = 0$