An Addition-Formula Function

A function $f(x)$ defined on $(-1, 1)$ satisfies:

• for all $x, y \in (-1, 1)$, $\ f(x) + f(y) = f\left(\dfrac{x + y}{1 + xy}\right)$;
• $f(x) > 0$ whenever $x > 0$.

Determine, with proof, which of the following are true:

1. $f(x)$ is odd; 2. $f(x)$ is decreasing; 3. if $f\left(\dfrac{1}{5}\right) = \dfrac{1}{2}$, then $f\left(\dfrac{1}{2}\right) - f\left(\dfrac{1}{7}\right) = 1$; 4. $f\left(\dfrac{1}{11}\right) + f\left(\dfrac{1}{19}\right) + f\left(\dfrac{1}{29}\right) + f\left(\dfrac{1}{41}\right) < f\left(\dfrac{1}{3}\right)$.