It occurred to me that there's one thing we may want to clarify for you Matias: There's a difference between the idea of an API to talk to Sandstorm (the platform) per se, vs an API to talk to specific Sandstorm *apps*. Jacob mentioned that some apps have APIs, such as Tiny Tiny RSS. However to answer your question, Sandstorm the platform afaik does *not* have an API. So that's a big part missing from your vision that doesn't exist and probably won't for a good while. (Jacob correct me if I'm wrong there, though). Indeed, Sandstorm was built for the web first.
But let's consider your example. It now makes a little more sense to me (vis a vis the router setup comparison):
* install Sandstorm on your server
* install "Sandstorm Mobile" on your phone
* Sandstorm Mobile: Log into your Sandstorm server
* Sandstorm Mobile: Install Dropbox Replacement onto your Sandstorm server
* Sandstorm Mobile: Configure Dropbox Replacement to start syncing files with your phone
Again, first major blocker is that there's no API for Sandstorm the platform. But the second issue that I'd put forth is, Sandstorm doesn't have a set of "official Sandstorm apps" as such, it's more open-ended in spirit. (In theory anybody could even make their own market). So, there might be multiple Dropbox Replacement apps some day. Thus, I don't think it would make sense to have one "Sandstorm Mobile" with special handling for all of of them, since each app's APIs would be different. Instead, the Dropbox Replacement developers can create their own Dropbox Replacement Mobile app, like Tiny Tiny RSS did.
So here's how I could see your idea working: Sandstorm Mobile could be a launcher of sorts. You use it to install your Dropbox Replacement app on the server. You use it to create a new instance ("grain") of Dropbox Replacement. Then, you tap to open the grain (still within Sandstorm Mobile), which *launches a different app*: the Dropbox Replacement Mobile app, and it connects to that grain. From there you can configure your shared folders etc. (Then you go back to Sandstorm Mobile, create a Tiny Tiny RSS grain, open it, and it launches the Tiny Tiny RSS mobile app, etc)
This is all *plausible* and I would say within the vision of Sandstorm, and I agree it's an appealing vision. But again, the prospects of that are quite a ways away. The current Sandstorm platform is something of a legacy thing. The "next generation" platform known as Tempest would be where such new features would be implemented. Tempest is in early stages still. Again, Sandstorm was built for the web first, so as it stands that's what it "naturally" is.