We are tasked with finding the derivative $f'(0)$ using the three properties of the function $f(x)$:

1. $f(0) = 0$
2. $\lim_{x \to 0} \frac{f(x)}{x} = 1$
3. $f(x + y) = f(x) + f(y) + x^2y + xy^2$

To find $f'(0)$, we will use the limit definition of the derivative:

$f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}$

Since $f(0) = 0$, this simplifies to:

$f'(0) = \lim_{h \to 0} \frac{f(h)}{h}$

From property (2), we know:

$\lim_{x \to 0} \frac{f(x)}{x} = 1$

Thus, using this result, we conclude:

$f'(0) = 1$

Therefore, the derivative of $f(x)$ at $x = 0$ is:

$f'(0) = 1$

Would you like further details on any step or concept used? Here are 5 questions that may deepen your understanding:

1. How does the h-definition of a derivative work in general?
2. What does the property $\lim_{x \to 0} \frac{f(x)}{x} = 1$ imply about the function near $x = 0$?
3. Why does the expression $f(x + y) = f(x) + f(y) + x^2y + xy^2$ help us analyze the structure of $f(x)$?
4. Can we infer anything about the higher-order derivatives of $f(x)$ from this?
5. How can we generalize this process to find $f'(a)$ for any $a$?

Tip: When solving limit-based problems, carefully examine how given conditions (like $\lim_{x \to 0} \frac{f(x)}{x} = 1$) connect with definitions of derivative.