a) If $d_1, d_2, \dots, d_p$ are the positive divisors of a positive integer $n$, then $\{d_1, d_2, \dots, d_p\} = \{\frac{n}{d_1}, \frac{n}{d_2}, \dots, \frac{n}{d_p}\}$.
Let $b_1, b_2, \dots, b_q$ be the positive divisors of $b$. Then
$$Q(a, b) \le \frac{a}{b_1} + \dots + \frac{a}{b_q} = \frac{a}{b}\left(\frac{b}{b_1} + \dots + \frac{b}{b_q}\right) = \frac{a}{b}S(b) = \frac{a}{b}Q(b, a), \quad (1)$$
therefore $\frac{Q(a,b)}{a} \le \frac{Q(b,a)}{b}$.

Since the statement is symmetrical with respect to $a$ and $b$, $\frac{Q(b,a)}{b} \le \frac{Q(a,b)}{a}$, hence $\frac{Q(b,a)}{b} = \frac{Q(a,b)}{a}$, which shows that (1) is an equality.
This shows that $a$ is divisible with all the divisors of $b$ and $b$ is divisible with all the divisors of $a$, so $a = b$.

b) It is not true. For instance, if $a = 2$ and $b = 5$, then $S(2) + S(5) = (1 + 2) + (1 + 5) = 9$ and $Q(2, 5) + Q(5, 2) = 2 + (5 + 2) = 9$, but $a \neq b$.