Newton's three laws are general properties of all forces. Newton's Law of Gravitation and Coulomb's law of electrostatics are specific properties of the gravitational and Coulomb force respectively.
Is it correct to assume that the proportionality of earth's gravitation exerted on objects of different mass or weight needs some other law that is not implied in Newton's three basic laws of motion?
Yes. This is a specific property of gravity. So it's not derive-able from the three laws. The Coulomb force does not have this property
If the above is correct, is g ("9.x m/s2") a derivation of Newton's general law of gravity? How exactly does this derivation look like?
You can't derive laws, because laws are not theorems. Laws are axioms. They have to be postulated, not derived.
You have to deduce laws by observing nature. Observing that things fall at the same rate regardless of their mass helps in the deduction.
By observing that objects of different masses have the same acceleration, you deduce that force exerted by Earth on the object $F_{EO}$ is proportional to the Object's mass $m_O$.
By symmetry, the force exerted by an object on Earth, $F_{OE}$ is proportional to $m_E$.
By Newton's third law, $|F_{OE}|=|F_{EO}|=F$. So, $F$ is proportional to both $m_E$ and $m_O$, i. e. $F$ is proportional to $m_E m_O$
The dependence on $\frac{1}{r^2}$ is not observable on falling objects, because $g$ does not vary much near the surface. You have to deduce the proportionality of $F$ to $\frac{1}{r^2}$ from planetary motion observations.