That's not what's going on. The space twin is younger than the earth twin because the space twin's path through spacetime between the separation event and the reunion event was shorter than the earth twin's path between those two events.

The acceleration is something of a red herring - it just so happens that in flat spacetime there is no way of setting the two twins on different paths through spacetime without accelerating one or the other. However, there is a completely acceleration-free version of the experiment in which the space twin turns around by doing a hyperbolic orbit around a distant star. In this variant neither twin experiences any acceleration, but their paths through spacetime are still different lengths and the one that takes the longer path will age more.

In GR it does. In SR it is somewhat arguable, because global inertial coordinates do have a "privileged" position in flat spacetime; but you can do SR in non-inertial coordinates in which an accelerated object is at rest (for example, Rindler coordinates for an object with constant proper acceleration) and get all of the same physical predictions. So the principle can indeed be applied to accelerated motion even in SR.

I'm not sure what you mean by "this". Global inertial coordinates do not exist in a curved spacetime, so what I said about SR and global inertial coordinates isn't even meaningful in GR to begin with.

I won't learn tensors for about a year, but when I look at SR written in tensor format, it looks a lot like GR.

So I was wondering if the SR way of handling an acceleration could be modified by adding curvature of spacetime and if doing so would that basically be a formulation of GR.

I'm not sure what this means either. But perhaps this will help: spacetime curvature in GR is defined using geodesics, i.e., the worldlines of freely falling objects. There is no need to even look at acceleration (meaning proper acceleration).

Or try this: One way of describing the difference between SR and GR is that, in SR, you can choose not to deal with curvilinear coordinates, because there are non-curvilinear (inertial) coordinates that cover the entire spacetime; but in GR, you have no choice, you have to deal with curvilinear coordinates, because there are no non-curvilinear coordinates that cover more than a small patch of spacetime. This is true regardless of the state of motion of the objects you're dealing with: in SR, you can use global inertial coordinates to describe accelerated objects; and in GR, you have to use curvilinear coordinates even if you're only interested in freely falling objects.

Perhaps: non-inertial frames (which would include any formulation of a rest frame for a particle undergoing proper acceleration) in SR do look very like GR. And handling non-inertial frames in flat spacetime gave Einstein clues towards developing GR.