To fill the space that is both above y and above y2, you can take the maximum of both. In order to only fill where y1 is above y2, you can you the where parameter:
plt.fill_between(x, np.maximum(y(z), y2(c)), y1(x), where=y2(c)<=y1(x), alpha=0.5)
You might need to use more than 100 points in the linspaces to avoid small regions without filling. For this image I used np.linspace(0, 12, 500).
To have nicer labels, you could you latex format (enclose with $ signs, and add braces for the powers):
plt.plot(..., label="$y=(18-x^2)^{1/2}$")
plt.plot(..., label='$y=3*2^{1/2} - (18-x^2)^{1/2}$')
To get a square root symbol, use the latex function \sqrt. To have a backslash in a Python string, either the backslash should be doubled, or the string should be preceded by an r (raw string).
plt.plot(..., label="$y=\\sqrt{18-x^2}$")
plt.plot(..., color='red', label='$y=3*\\sqrt{2} - \\sqrt{18-x^2}$')
