The maximum distance of gravitational attraction I'd like to better understand how gravity functions at extreme distances.  Here's what I'm struggling with:

*

*In Newtonian mechanics, every mass in the universe attracts every other mass inversely proportional to the square of their distance from each other.

*In quantum mechanics, space and energy are discrete.

*The Universe is expanding, and the expansion is accelerating over time.

So imagine a universe with only two entities.  As I place those entities further and further apart, the attraction between them grows vanishingly small, but never reaches zero, according to #1 above.
But at some specific distance, the attraction between them would be so small that they wouldn't move a single (#2) discrete distance (plank length?) toward each other in a year.  They wouldn't gain a single quanta of energy in that time.  So in what sense is there an attraction between them at that distance? Is it possible to slowly accumulate a single quanta of energy over an arbitrarily long period of time?
I'm obviously misunderstanding something fundamental here!
And at some distance, #3 suggests that the expansion of space between the two bodies will overwhelm the gravitational attraction?  Is there some formula to describe the maximum effective range of gravity given the masses of the two objects?
 A: Bear in mind, that I hadn't have single course of general relativity, only special.
So first, lets go over your statements:

*

*True

*Debatable or even not true. As mentioned by @G.Smith in comment, free particle is a good example. Energy levels are generally discrete in bound systems.

*Debatable. Observations suggest, that Universe is expanding, but from purely theoretical point of view, it could also be stationary or shrinking.

So now some general ideas. Newton Physics + Maxwell equations are well observed in real life, on a scale from crystals to planets. BUT it does NOT describe structure bellow the atom (or even between molecules) in some cases. We have experimental proofs for that. It also fails on the scales of Galaxies, when relativistic effect take place.
Quantum mechanic is very useful and work on small scales, but it is hard to use on scales bigger than crystals. If we use relativistic quantum mechanic it can als describe wery high energy particles. But it DOES NOT describe gravity at all. If we have system, that includes gravity we can't use QM.
General relativity takes into account effect of mass on "space-time". It is useful to describe gravitational interaction between very big objects. But bear in mind, that general relativity is a demon from mathematical underworld and very hard to either understand or use. Special relativity on the other hand only describes very fas (high energy) objects, which space bodies, generally aren't.
Expanding of universe is purely general relativity phenomena, and cannot be described neither with QM or Newton's physics.
Booth QM and relativity have somethink called "classical limit". This means, that if we describe something of medium size, eg. average experimetn from Newton's physics they give the answer which is so close to Newtonians prediction, that the error is smaller than can be observed.
BUT QM still doesn't tell anything about gravity. And general relativity doesn't describe anything on subatom levels. So generlay you are right. There is something wrong. This 2 theories (QM and general relativity) aren't compatible with each-other. This simply means that either one of them needs to be improved, or that they are booth only special cases of some more general theory eh. "Grand Unified Theory", which still need's to be "invented".
So generaly, your misunderstanding comes from using "special" theories for fields of physics, where we know, that they don't work.
A: So, like, your basic problem, if I tried to get rid of everything else I want to comment on, has to do with what you are trying to put together.
(1) is a statement of Newtonian gravity, (2) is possibly a statement of loop quantum gravity or so [it is a stronger statement than quantum mechanics would make], and (3) is a statement of general relativity. They are different models of the universe, of course they're going to say different things about it.
Of these probably (2) is irrelevant to what you are asking about and (1) and (3) can be reconciled in general relativity, so I recommend you ask purely about that.
In general relativity, yes, most of the galaxies appear to be moving away from us except for a couple of galaxies nearby us which are not so limited, and in fact the Andromeda galaxy is going to collide with the Milky Way galaxy eventually. So there is a sort of distance scale where these two factors described by general relativity can balance out, a “change of regime”, and it is experimentally at intergalactic distance given galactic masses.
I suppose there is some amount of massaging the FLRW metric where the scale factor can give you a distance relationship for the exact cutoff, but this gets complicated by the fact that the expansion is speeding up, which requires a different model, usually now the ΛCDM model. I don't think that you can get a perfect balance in ΛCDM but I am mostly trained in condensed matter physics so this goes way beyond my comfort zone.
