I read somewhere that action-reaction forces satisfy the form
$$ \mathbf{F}_{12}=-\mathbf{F}_{21}=f(|P_1-P_2|)(P_1-P_2) $$
meaning that the modulus can only depend on the distance between the two points $P_1$ and $P_2$.
Is that right?
Cannot the modulus depend on, say, $|\mathbf{v}_1-\mathbf{v}_2|$?
Yes, in principle it may depend!
In pure Newtonian mechanics (not Lagrangian), invariance under the Galileo group requires that for a pair of isolated points $P_1,P_2$ described within an inertial reference frame, the force on $P_1$ due to $P_2$ has the general form: $$\vec{F}_{12}= \vec{F}_{12}(P_1-P_2, \vec{v}_1- \vec{v}_2)$$ with $$\vec{F}_{12}(R\vec{x}, R\vec{u})= R\vec{F}_{12}(\vec{x},\vec{u})\quad \mbox{ for all $R\in O(3)$.} $$ Conservation of momentum implies $\vec{F}_{12}= -\vec{F}_{21}$. Alternatively this constraint can be imposed as the action-reaction principle obtaining the conservation of total momentum.
If you require also the conservation of total angular momentum you also have to impose that $\vec{F}_{12}$ is directed along the segment joining $P_1$ and $P_2$. Summing up you get, $$\vec{F}_{12} = f\left(|P_1-P_2|, |\vec{v}_1- \vec{v}_2|, \alpha(\vec{P_1P_2},\vec{v}_1- \vec{v}_2) \right)\: \vec{P_1P_2}$$ where $f$ is a scalar function and $\alpha(\vec{u},\vec{v})$ is the angle between $\vec{u}$ and $\vec{v}$.
The fact that $\vec{F}_{12}$ is directed along the segment joining $P_1$ and $P_2$ is sometime called stronger form of the action-reaction principle.
Finally, dealing with a set of $N>2$ points, $P_1,\ldots,P_N$, the superposition principle requires that the total force acting on a point, say $P_i$, is the sum of all forces acting on $P_i$ due to $P_j$ with $j\neq i$ as if that pair were isolated.
ADDENDUM. In principle $\vec{F}$ could depend on higher derivatives of the positions than velocities (e.g. accelerations and velocity of accelerations as it happens in the semi classical model of the electron). However as soon as one allows it, there is no longer guarantee for the so-called determinism, the fact that, under suitable initial conditions (positions and velocities in the standard case), the evolution of the system is determined.
