What if the Sun were charged?

There's a reason I posed this analogy.

E&M is a well-developed and mathematically and physically consistent theory.

Here we have a paradox, and we can bring the paradox entirely within established theory to study it.

Step 1 is to thoroughly understand the distinction between force-from-static-charge and wave-signals-from-wiggling-charges, entirely within the context of E&M.

Step 2 is to examine how gravity might behave differently from static charge.

That way we don't get two sources of confusion at once.

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Gravitation and electrodynamics do have similar behavior.
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I agree.
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Their forces propagate at FTL speeds.
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I totally disagree. The force on my test charge is caused by its interaction with the local field. The direction of that force depends on the shape of the local field. The shape of the local field depends on the history of the surrounding field, and so on, back to the boundary conditions. One of the boundary conditions is the position of the sun 8 minutes ago. This is the ocean wave analogy, and it's a good analogy. Electric forces are governed by the diffusion equation. Force is the derivative of potential with respect to distance. The direction of the force is in the direction of the gradient of the wave--the steepest ascent of the wave. The magnitude of the force is the slope of the wave. The height of the wave is the voltage. Remember, I'm talking about E&M because I don't want an extra source of confusion.

In the case of a uniformly moving charged sun, the ocean waves become a cone centered at the sun (magnitude proportional to inverse r).

The opposing view says, I suppose, that the electric force is like a series of little balls that are thrown from the sender to the receiver (or a hole where the ball would have been but got blocked), and the direction of the force is the direction of travel of the ball. OK. It's a model. I can't argue with it except that it gives the wrong results. (Well, I could point out that electric fields do not necessarily go back to a source, but can go in circles.) So now we need a new postulate: that the balls travel immensely fast.

What other reason is there to postulate this, other than to prop up a model which gives different results than the established ocean-wave model? (And remember, I'm still talking about the electric force.)

It is true, but irrelevant, that photons can travel faster than c. See Feynman's book "QED: The Strange Theory of Light and Matter." It is irrelevant because for paths faster or slower than c, nearby paths have greatly different phases and so they cancel. Only at c do nearby paths have similar phases and so they add.

Since gravitation and electrodynamics have similar behavior, do we also need a "Pushing and Pulling Electricity" model?

Conversely, if we don't need such a model for electricity, why would we need it for gravity--since gravitation and electrodynamics have similar behavior?
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And when something disturbs the associated potential fields, those waves travel at speed c in both cases.
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I agree.

If there is no "action at a distance", then what is there?

If two objects exert a gravitational -- or electrical -- force on each other, but there is a time lag, then how is momentum conserved?

Worse, if (as SR tells us) there is no simultaneity, so that a movement of object A could be "simultaneous" with any of a number of movements of object B, then how could their changes of momentum be balanced?

It's very simple: Action is local, and momentum is conserved locally.

The charged object interacts with the local electric field and changes its momentum. In so doing it is accelerated. This produces a bend in the E field, which propagates outwards (it's our friend diffusion again). This bend, this disturbance, carries with it momentum equal to the momentum change of the object.