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[MadMojoMonkey]

Is the effect of gravity asymptotic with distance? For example does a single proton floating around exert a tiny effect on space on the other side of the galaxy?

Yes. gravity is space-time curvature, and everything effects everything.

That said. The intensity of the interaction between 2 masses diminishes proportional to the square of the distance between them.
F = GMm/r^2
It's the {...}/r^2 part I'm talking about.

Suppose: 2 objects are some distance apart, call it {ong}, and they express an equal and opposite force on each other of {bonga}.
If all else is the same, but the distance between the 2 objects is doubled to 2*{ong}, then the force they NOW express on each other is {bonga}/4.

Over very long distances, the force expressed is so tiny as to be nearly non-existent except when applied over vast distances for a vast time period.

Ultimately, theory states that the space-time curvature caused by any mass has no limit to spacial influence, given infinite time for the gravity to propagate the distance.
I mean. The pull of gravity of every object in the universe extends to every other object in the universe. Gravity moves at the speed of light, so it takes time for changes to be 'felt' by distant observers.

It seems to me like gravity can't know what constitutes a contiguous mass-energy, so it must be just the accumulation of the effects of every particle in the vicinity.

Exactly. You have excellent word-choice here.

(Even though space-time curvature can't "know" anything. It can be quite hard, even for physicists, to avoid anthropomorphic terms when describing things.)

And if the particles in my body are exerting a tiny amount of gravitation to matter in Andromeda 2.5 million light-years away, is this effect instantaneous? These seem like questions that can't really be answered, but I'm curious where the math/theory points.

Changes in space-time curvature propagate at the speed of light, according to the Einstein Field Equations.