What we care about is the effect of gravity on an object, and what we care about is the change in the momentum of the object. So we set m to keep the speed constant. (You can think of m as a powered object that keeps v unchanged).

I understood v as an initial impulse, propelling m radially, which is nullified by g after a short time interval.

Given this new requirement, you add an acceleration a, equal to -g.

x=x_0 +vt +.5at^2 -.5gt^2

In the graphic, x0=0.

Path p1 (green) is with an initial impulse of v.

Path p2 (black) is with an acceleration of a=-g.

P2 is an inertial path (constant velocity) with NO net forces acting upon m.

There is no change of mometum.

1. No matter how fast the object

is, gravity will act on the object instantaneously.

...

2. In a very short time slice dt, we can assume that m is stationary and the gravity received

is constant,

1. Because the gravitational energy from M, moving at c, has conditioned the surrounding space before the object m has reached any specific location.

2. Not unless you want to rehash Zeno's motion paradox, which is solved by 'the motion is constant', thus cannot be zero.