Correlating Newtonian Model with Einstein's GR

Well, this is your view, how do you look at it.
No, that isn't my view, that's what the law states.

These are standard physics terms already well defined. So new definition is necessary. Here A can be considered as area.
"stress" is not a notation I'm familiar with, but I see that it is used on Wikipedia too:
https://en.wikipedia.org/wiki/Stress–strain_curve

One problem is that you cannot have a strain on a particle; a particle's length is by definition so small its irrelevant.

But the real issue is that this is just some generic definition of stress, not something specific to this situation. Please give the equation and its derivation of this "Lorentz-stress", where the velocity of the particle is used as an input.

If it is not a straight line, as per Newton's Law of Inertia, it will be subject to a force.
Irrelevant. Do you agree with me that a not-straight line isn't necessarily a loop, yes or no?

"Equivalent of a straight line" and "a straight line" does not mean the samething.
Technically, you are right, but that's really not the point here. A geodesic in GR is what a straight line is in flat space. In that regard, they are the same thing.

It is the thumb rule, which bends magnetic field lines around a current carrying wire.
The "thumb rule" is not a force, so that's wrong. Please give a force.

"can be considered as" and "is" are not the same thing. As I already said, that's an imperfect analogue, so it will break down in places.

You have described it literally as quantization:

How will you define continuity of a function? Say consider f(x)=x. Is it continuous or quantized?
This is a very basic calculus question. Off the top of my head: if a function has a well-defined derivative on its entire domain, it's continuous on that entire domain. f'(x)=1, so yes, that's continuous.

A quantized (or as it's called in mathematics, "discreet") function will not be continuous, in general.

Well, Hubble observed expansion from the Earth.
No, Hubble observed galaxies moving away from us, not the expansion directly. You've missed the point: rulers stretch too, so against what ruler are you measuring this expansion? I'm not saying it's impossible, but one has to be very carefully to define these things properly. The rubber sheet analogue doesn't work properly in these cases.

Why they will remain stationary with the rubber sheet? No force is applied to the balls.
First of all, there's Newton's First Law. But in reality, it's more complicated than that.

Pick a ball. The sheet is being stretch in all directions, out from underneath this ball. There is no force pushing the ball around; it remains stationary compared to the sheet directly underneath. The ball doesn't move.
Pick another ball. The sheet is being stretch in all directions, out from underneath this ball. There is no force pushing the ball around; it remains stationary compared to the sheet directly underneath. The ball doesn't move.
Both balls don't move with respect to the sheet underneath, because there's no force. Yet, the distance between the balls increases!

"The mechanical energy can be expressed as follows, using the notation of Carroll and Ostlie:

The above expression, I copy pasted from that link.
OK, so the first link you gave was indeed wrong.

Mechanical energy isn't the same as work, so the quoted part doesn't talk about work either.