Question: Given a collection of point-particles, interacting through an attractive force $\sim \frac{1}{r^2}$.

Knowing only $m\_1a=\sum\_i \frac{Gm\_1m\_i}{r^2}$ and initial conditions we can deduce the motion of the system.

Consequently we can observe that three quantities remains constant
A) center of mass of the system
B) total energy
C) angular momentum

How can we derive these 3 facts directly from $m\_1a=\frac{Gm\_1m\_2}{r^2}$ ?

Are these quantities conserved for any attractive force $\sim\frac{1}{r^n}$ ?

Given any monotonically decreasing force for $r$ in $(0,\infty)$, which are the conserved quantities?

Answer: 
These quantities will be conserved by any system whose laws respect translational and rotational symmetry and are static in time. Respecting translational symmetry means that interactions don't depend on the positions of individual particles, just on differences of those positions. Respecting rotational invariance in turn means that interactions depend only on the magnitude of those differences. Being static means that the interactions don't depend explicitly on time or velocities. So this means that the two-point interaction must look like $V({\mathbf x}, {\mathbf y}) = f(|{\mathbf x - \mathbf y}|)$ (with $f$ suitably smooth). So yes, in particular your $r^{-n}$ potential will work.

Note that in general you could also consider more general interactions e.g. three- or more-point interactions. But to conserve the said quantities the interactions must still be singlet states of the respective groups.
All codes of the N-body simulation make use of instantaneous propagation of the speed of gravity.
The other answers follow the same criterion that is good enough for local simulations.

**As a consequence of the finite speed of propagation of gravity and the motion of the masses there is a dissipative component.**

As the bodies are never static, the masses continuously lose energy to space because there is no known mechanism that reverses the direction of flow.

(An equivalent effect applies to the radiation of charged particles by virtue of its continued accelerated motion.)

from **Jorgen Kalckar and Ole Ulfbeck experiment** (a) (energy of two masses and a spring):
...

> 
> The difference between these two
>  energies is lost by each mass: it is
>  taken away by space-time, in other
>  words, it is radiated away as
>  gravitational radiation
> 
> 
> gravity can behave like a wave:
>  gravity can radiate
>  ...
>  All this follows from the expression of universal gravity when applied to moving observers, with the requirement that neither observers
>  nor energy can move faster than c.
> 
> 
> 

(a) ***motionmountain*** free online book, ch 18, pag 526/527: Motion in General Relativity, Gravitational Waves.