Curvature affects motion by making the lines that are as straight as possible end up converging, just line how if you and your friends fly at constant altitude from the north pole, then no matter what directions you go (even if you and your friend head out in very different directions) then you start to converge on the south pole. This is a very good way to describe an effect that is determined by the path and not by the mass of the object taking the path. This is sometimes described as "spacetime tells matter how to move" but really this is just that the straightest possible lines converge when spacetime is curved the right way.
How does this relate to those worldlines near a planet? They try to move in the straightest possible lines, and to first approximation everything is exactly as in SR, so you are right there, but that isn't gravity or curvature, that's just that curved things don't look curved when you zoom in. So if you were Newton you'd say that things get pulled to the center of the planet so things of a certain size would start to move closer to each other as they each fell towards the same center of the planet. That's what curvature does, and it does it in a way where the path doesn't depend on whether a light or a heavy particle does it.
But I'll say this, that isn't gravity. That's just curvature. And while the commonest effects of gravity are actually the effects of curvature, they are different things. Let's look first at curvature some more to see what causes it and then we can compare that to gravity very carefully. Something that is usually not mentioned enough is that while mass, energy, momentum, stress, and pressure are sources of curvature, they are not the only things that create curvature, curvature itself can create further and additional curvature. A gravitational wave can propagate or even spread in a vacuum of empty space devoid of all mass, energy, momentum, stress, and pressure.
The region outside a symmetric nonrotating static star is curved, even the parts far from any mass or energy or momentum or stress or pressure. The space remains curved because the existing curvature is exactly shaped so as to persist (or otherwise cause future curvature exactly like itself).
So curvature allows and sometimes requires more and/or future curvature, just as a travelling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later. The vacuum allows curvature far from gravitational sources just as it allows electromagnetic waves far from electromagnetic sources. What electromagnetic sources allow is for electromagnetic fields to behave differently (namely to gain or lose energy as well as move in different ways and gain and lose momentum and stress). Similarly what gravitational sources do is allow curvature to react differently to itself than it otherwise would.
Imagine a flat region of space shaped like a ball, then imagine a funnel type curved space where two regions of surface area are farther apart than they would be if flat (like a higher dimensional version of a funnel and on a funnel surface two circles of a particular circumference are farther away as measured along the funnel then if two similarly sized circles were in a flat sheet). On its own, spacetime doesn't allow itself to connect those two kinds of regions together, but that mismatch is exactly the kind or not-lining-up that putting some mass or energy right there on the boundary fixes. So without mass those two regions can't line up, with mass they can. Just like an electromagnetic field can have a kink if there is a charge there.
So your curvature likes to propagate a certain way, and if you want it to deviate from that, you need mass, energy, momentum, stress, and/or pressure. And you'd need the right kind to get it to match up, the kind you want might be available, and might not even exist, so not all kinds of curvature will be allowed. But the point of a source is that it changes the balance between nearby curvature and not that affects future curvature. So there is a kind of balance, and there are things that can warp that a balance. Those things that warp that natural vacuum balance are called gravitational sources.
Having curved spacetime is something we observe. Having gravitational sources that can change the normal or usual way curvature evolves is something else entirely. We can make theories about how the sources evolve, and then the curvature is forced to co-evolve with it, and that's what gravity is about, about gravitational interactions (source and curvature together) changing how the curvature evolves changing the evolution that the curvature otherwise would have evolved a different way.
So there is nothing circular, curvature is observed, and on its own it interacts and affects itself in a particular way (that is also observed), but gravitational sources get to change that and by interacting with the gravitational sources (which we can do) we can ourselves make the curvature change in different ways than it otherwise would!
So now let's get to your questions:
If space-time curvature causes objects to fall, how?
Spacetime curvature (if in vacuum, and sometimes even in some kinds of matter) makes paths that are straight as possible converge. And that, not falling is what gravity does. Objects on the surface of the earth are stressed and are pushing to resist that stress and that's why you feel the floor push up on you which is why you don't move inertially, you don't fall.
Is space moving so as to push or rotate mater? No, but matter does expand and push when it is under pressure, that;s why you feel the floor pushing up on you.
How does something about the mass energy tensor alter geodesics or 4-velocity vectors?
It does not alter geodesics. You move along geodesics because spacetime is curved. Spacetime is curved differently than it would be all on its own because the sources allow more novel arrangement.
Curvature is one thing, and there is one way it evolves. When there is are gravitational sources, even more manifold possibilities are available. Gravity is about gravitational sources, which allows different curvatures. The curvatures are what is observed. Keep gravity, and the effect of sources on curvature, separate from the curvature itself which can influence itself. It's like keeping in mind the difference between an electric charge and an electric field.