I liked Batiatus' answer where we distinguish between flat spacetimes and curved ones, I feel like that gives you a good answer of questions (1) and (2). To answer question (3) I would question this premise that special relativity doesn't “do” acceleration, because I believe that that's the only thing it does.
So a remarkable thing to my mind is that the Lorentz transform can be reconstructed as the limit of itself truncated to first order,
$$
\lim_{N\to\infty} \begin{bmatrix}
1&-\alpha/N&0&0\\
-\alpha/N&1&0&0\\
0&0&1&0\\
0&0&0&1
\end{bmatrix}^N=\begin{bmatrix}
\cosh\alpha&-\sinh\alpha&0&0\\
-\sinh\alpha&\cosh\alpha&0&0\\
0&0&1&0\\
0&0&0&1
\end{bmatrix}.$$
In a way, this means that what special relativity is “known for”—time dilation and length contraction—are not what it is really “about.” Those are second order effects that come directly from the first order effect, they are consequences rather than new physics in themselves. The new physics is the relativity of simultaneity, $\gamma$ is a footnote, a normalization that Nature happens to perform for us to keep the relativity of simultaneity mathematically consistent.
Furthermore these first order boosts happen directly when we accelerate. What it is saying is that when you accelerate, clocks in front of you by a coordinate $x$ appear to tick anomalously faster (than say Doppler would suggest), and clocks behind you slower, by a factor of $1+ax/c^2$. That is the new physics we are talking about here in special relativity.
The choice to use differential geometry to describe the situation is orthogonal. You can phrase the above as Rindler coordinates if you like, using differential geometry in special relativity. But of course you do not have to. You can also just calculate directly, for example, that there is an event horizon at coordinate $x=-c^2/a$. After all I just told you that the clocks aren't ticking there. But if you want to see that event horizon in Rindler coordinates, don't let me stop you!
Similarly, while differential geometry has been the most successful approach to general relativity and I wouldn't want to supplant it, it is not necessary in order to get your bearings there. At least, not in the case of static situations The essential static physics of general relativity is the equivalence principle, which combines with special relativity in a pretty straightforward way. If you wanted to predict that gravitational waves exist but require a varying quadrupole moment of mass you might require more than just the equivalence principle, you might require the full Einstein equations and differential geometry.
But the equivalence principle is enough to conclude that GPS satellites will be seen by us on the ground, who by the equivalence principle are accelerating toward space, as ticking faster than us—real gravitational time dilation. And it is enough to conclude that there may be places in the universe which are black holes, that event horizon that we saw before becoming an essential part of a location in space via the equivalence principle. We just don't see it because our acceleration $g$ satisfies $c/g\approx0.97\text{ years}$ and so this wall of death should be about a light year underneath our feet, but Earth is much smaller than a light year. You can understand a bit without invoking the full differential geometry approach.
So in some sense, acceleration is the only thing that special relativity does, and you can always get away with using special relativity to describe acceleration. Indeed, to my mind the resistance to talking about acceleration is why people think the twin paradox is a thing. If you know that this is the core of the physics, the twin in space is accelerating towards their sibling on Earth which is a long distance away, so they see the Earth twin age extremely fast. So much for the paradox.
