Confusion about gravity I understand the “rubber sheet” model of Relativistic gravity is just an illustration, and beyond the initial issues of mixing three-dimensional objects with a two-dimensional representation of 3D space/time, the basic concept is easy enough to grasp… The billiard ball “falls” into the depression of warped space/time created by the heavier, denser bowling ball, and is captured in orbit.
Let’s suppose, on the other hand, that rather than two dissimilar “balls”, we place two identical, inert, non-radiant particles (dark matter?) into the model, let’s say 100,00 light years separating them, and get rid of the bowling and billiard balls altogether. Each particle warps the spacetime fabric equally, and since we imparted neither directional motion nor rotational spin to either particle, and there is no emission of energy/light to create pressure, it would seem that, according to Einstein, neither particle would move from the spot we placed it, as there is no “slope” to “fall” along when both particles are deforming the sheet equally.
But according to Newton/Galileo, both particles would instantly, (not 100,000 years from that point), be drawn towards each other by the force of gravity, slowly at first, but with ever increasing acceleration. So, do the particles move towards each other instantly or don’t they? Who is right, E or N? Why?
 A: 
Each particle warps the space/time fabric equally, and since we imparted neither directional motion nor rotational spin to either particle, and there is no emission of energy/light to create pressure, it would seem that, according to Einstein, neither particle would move from the spot we placed it, as there is no “slope” to “fall” along when both particles are deforming the sheet equally.

The image of the rubber surface and distortions applies to the case of a much larger mass in the center, and a small one around it:

the Schwarzschild solution, it is assumed that the larger mass M is stationary and it alone determines the gravitational field (i.e., the geometry of space-time) and, hence, the lesser mass m follows a geodesic path through that fixed space-time. This is a reasonable approximation for photons and the orbit of Mercury, which is roughly 6 million times lighter than the Sun. However, it is inadequate for binary stars, in which the masses may be of similar magnitude.
The metric for the case of two comparable masses cannot be solved in closed form and therefore one has to resort to approximation techniques such as the post-Newtonian approximation or numerical approximations.

so it is not that simple to visualize:

The post-Newtonian expansion is a calculational method that provides a series of ever more accurate solutions to a given problem. The method is iterative; an initial solution for particle motions is used to calculate the gravitational fields; from these derived fields, new particle motions can be calculated, from which even more accurate estimates of the fields can be computed, and so on. This approach is called "post-Newtonian" because the Newtonian solution for the particle orbits is often used as the initial solution.
When this method is applied to the two-body problem without restriction on their masses, the result is remarkably simple. To the lowest order, the relative motion of the two particles is equivalent to the motion of an infinitesimal particle in the field of their combined masses. In other words, the Schwarzschild solution can be applied, provided that the M + m is used in place of M in the formulae for the Schwarzschild radius rs and the precession angle per revolution δφ.

It can be shown that Newtonian physics emerges at the limit of validity of General Relativity,
You state:

But according to Newton/Galileo, both particles would instantly, (not 100,000 years from that point),

General Relativity was partially invented to reconcile the velocity of light as a limit in special relativity and this assumption of Newtonian physics , which is wrong.
From the link:

That means that Newton's theory depends upon a notion of absolute simultaneity. A change there is felt here at the same moment. However Einstein's 1905 theory had banished absolute simultaneity from physics. Different observers would judge different pairs of events to be simultaneous. Newton's theory had to be adjusted to accommodate this new relativity.
The change needed was, apparently, straightforward. In the revised theory, a change in the sun should not be felt here on earth instantly, but only after a time lag of around 8 1/3 minutes, the approximate time light takes to propagate from the sun to the earth. Then absolute simultaneity would no longer be needed in the theory.
This meant that Newton's theory needed to be adjusted to look more like electrodynamics. In the latter theory, effects do not propagate instantly in the electromagnetic field; they propagate in waves at the speed of light.

etc in the link.
You state:

be drawn towards each other by the force of gravity, slowly at first, but with ever increasing acceleration. So, do the particles move towards each other or don’t they? Who is right, E or N? Why?

As it has been shown that Newtonian gravity emerges from the limiting case of General Relativity I trust that the numerical approximations needed for the setup you have visualized will give as a solution the correct Newtonian attraction between two massive bodies.
A: Assuming we are able to create the two bodies out of no where, and in the state that you described. (even though, I do not think it is possible to do so).
Not only this way, but it is next to impossible to measure speed of gravity in any way. We have been able to measure speed of gravitational waves, but not that of gravity. Doing that, just eludes scientists due to us not being able to create stuff out of nothing, or make it disappear totally. I am not aware of any experiment confirming speed of gravity (which can be different from speed of gravitational waves).
Therefore, I can not say who is right Newton, or Einstein. However, quantitatively, GR is much more accurate in pretty much any circumstances. Neither of them proves which one is right when it comes to speed of gravity. GR comes close, at least it predicted speed of GW and they have been confirmed.
To your slope part, per GR - 


*

*As soon as the particles came into being, the space, and the particles will start interacting due to certain properties of space. 

*Due to this interaction, the space will start to curve. Rubber sheet analogy, is not accurate, but just ok for describing..

*This curving effect will spread in spherical directions (3D) at the speed of light. It will spread in symmetric manner. i.e. no slope. You can imagine it as a dip of space is spreading from the particles.

*After 50K years, the two curving effects (dips) will meet one another, making it a slope. Because, now the curving effect starts to add here on and is no more symmetric for individual bodies. 

*After another 50K years, both the bodies will come to know that there is a slope, and will start moving towards one another.
