Most yeasts will take the cider all the way down to about 0.992-0.994. That's not an exaggeration. That is very dry. But often it takes a couple of weeks to get from 0.999 to 0.992 or whatever. It's those last few points that take forever. There's enough yeast for natural carbonation at that point; HOWEVER.......
If you want a sweeter cider than that, you either need to kill the yeast with sorbate and sulfite, then backsweeten, or you need to stall the fermentation with gelatin, racking, low temperature (that's what I do), or you can try bottling then heating the bottles to pasteurize although this seems dangerous to me and will probably make the cider taste cooked.
In any of these cases, natural carbonation is very difficult. I mean, if you want to carbonate a dry cider at 0.992, no problem, you can prime and bottle just like a beer, and it will ferment out back to dryness around 0.992. Just realize that in this case, it takes a good month or two to carbonate -- it's just a bit slower than it would be for beer, but it will work. But if you are killing your yeast, then obviously they won't do a very good job of natural carbonation anymore.
With my method of gelatin, racking, and chilling to stall the yeast before they're finished, basically waiting several months for them to get really really tired before bottling, I find that I can get carbonation after priming and bottling, but it takes a good 5 or 6 months at least for that to happen. In the meantime, I drink most of the cider while young, without carbonation, and it's only the last handful of bottles that eventually get sparkling, and then I drink those and it's gone. Works for me. But this requires maximum laziness I mean patience. And I am a very very lazy I mean patient guy.
You could of course bottle with forced carbonation if you have the equipment for that. That's the easiest, if you have the equipment (I don't).
So all in all, trying to get a sweet cider that is also carbonated is really a crapshoot, unless you force carb.