Intertemporal Utility Maximization
What you have is basically the Fisher Two-Period Optimization Problem. (Fisher)

For iii)
You first need to find the Euler equation, which tells you how to optimally trade off first-period and second-period consumption.

For starters, your optimization problem is set up incorrectly: 
$$\frac{C_{1}^{1-\theta }-1}{1-\theta }+\frac{1}{1+\rho}\frac{C_{2}^{1-\theta }-1}{1-\theta }+\lambda (w-C_{1}-\frac{C_{2}}{1+r})$$

You find the Euler equation by deriving and equating the first-order conditions of both consumption terms.

$$\frac{1}{1+r} C_{1}^{-\theta }= \frac{1}{1+\rho}C_{2}^{-\theta }$$

Then just solve for $C_{1}$, substitute $C_{2}=s(1+r)$, and plug into $s=w-C_{1}$, which gives:

$$
s=w-(1+r)^{\frac{\theta-1}{\theta}}(1+\rho)^{\frac{1}{\theta}}s \\
s=\frac{w}{1+(1+r)^{\frac{\theta-1}{\theta}}(1+\rho)^{\frac{1}{\theta}}}
$$

The derivative, giving the response to a change in $r$, is:
$$
\frac{\partial s}{\partial r}= -\frac{(\theta -1) w ( 1+\rho)^{1/\theta } (1+r)^{1/\theta }}{\theta  \left((\rho +1)^{1/\theta }+r (\rho +1)^{1/\theta }+(r+1)^{1/\theta }\right)^2}
$$

The derivative for $0&lt;\theta&lt;1$ is positive, indicating that a higher return on saving will lead to an increase in the amount saved. Preference for first period consumption is still low enough, so that the consumer is willing to save more to consume it later. Here the substitution effect dominates the income effect.

When $\theta&gt;1$, the derivative is negative, meaning that preference for first period consumption is so high that the consumer is willing to lower savings for current period consumption. Here the income effect dominates the substitution effect.