I am not a dev, so I may not get this perfectly right from a frameworks point of view, but...
Most of what you said above looks correct. I don't think OmniSoftwareUpdate works on iOS.
OmniPresence is an API for syncing documents to a WebDAV server. Here's our marketing page about it:
http://www.omnigroup.com/omnipresence/
So on the Mac, any app can just save files to a particular folder, and the OmniPresence client app takes care of the syncing.
On iOS, where apps can't (easily) see each other's files, you would need to do the syncing in your app, with OmniFileStore and friends.
Here's the commit where OmniDocumentStore and OmniDAV were peeled out of OmniFileStore:
https://github.com/omnigroup/OmniGro...3396fd282f9382 That might give a little insight.
Hopefully someone who actually worked on this/uses it daily will chime in.