Why is it that when two physical quantities are directly proportional to a third one, we multiply the 1st with the 2nd, equaling the 3rd? (F=ma) Why do we not, for instance, add them?
How can we possibly figure out if such relations contain square roots, variables raised to powers, or any other specific mathematical operations?
Well, this is exactly what directly proportional, as you said, means.
It's easy to construct an experiment to show that $F = m + a$ is indeed often wrong.
For example in Kepler's second law:
$$\frac{P^2}{a^3} = C$$ (where $C \approx \frac{4 \pi}{GM}$)
These relationships are discovered through careful experiments, noting the data, and finding mathematical descriptions that fit. Usually, often, we also try to find a motivation for why the relationship looks like this.