How can the required wing area for climb be calculated, given aspect ratio?
Let's start with 3.19, since you need the result for 3.20. The parameters given suggest that we should calculate drag $D$ with the parabolic formula $$D = \frac{\rho}{2}\cdot v^2\cdot S\cdot\left(c_{D0} + \frac{c_L^2}{\pi\cdot AR\cdot e} \right)$$
I haven't seen the symbol $R_A$ before, but its magnitude suggests it is the aspect ratio (normally $AR$ in the US, $\Lambda$ elsewhere). Since the whole is an optimisation problem (we look for flight with minimum power consumption), we can determine the lift coefficient $c_L$ already. For propeller aircraft the point of minimum power is when the lift coefficient is $$c_L = \sqrt{3 \cdot \pi \cdot AR \cdot e\cdot c_{D0}}$$
Hey, you know all terms already! This allows you to proceed to the wing area once the needed lift is known. My next heroic assumption is that $W$ actually denotes a mass (the unit seems to say it is pound-force). The power plant should weigh 100 - 150 lbs already, so 200 lbs for the whole aircraft seems reasonable. So you can proceed:
$$L = 200 lbs \cdot g = \frac{\rho}{2}\cdot v^2\cdot S\cdot c_L$$
Plug in what we know, use sea level density ($\rho = 1.225 \frac{kg}{m^3}$) and the previously calculated lift coefficient:
$$S\cdot v^2 = 1527 \frac{m^4}{s^2}$$
This gives you pairs of wing area and speed squared which together fulfil the requirement. Some pairs, though, make more sense than others. Generally, the biggest wing will need the lowest power but will become increasingly fragile as wing area increases.

But there is still one requirement left: We cannot use more power than those 0.33HP:
$$P = 0.33\cdot0.7 HP = 172 W = D\cdot v = S\cdot v^3\cdot\frac{\rho}{2}\cdot0.04$$
$$\rightarrow S\cdot v^3 = 7094.8 \frac{m^5}{s^3}$$

Both conditions can be fulfilled when the flight speed $v$ is 4.646 m/s. This makes the wing area $S$&nbsp;=&nbsp;70.735 m².

Solving this for Denver, CO and finding the climb speed with 0.6 HP of power should be easy now. Just calculate the SEP and you know the climb speed.