Sharepoint has a Document Management Programming Model. From what I understand, this is used to create a document management environment and will allow users to access, share and add/edit documents based on the rights they have.
The questions, is there an API that can allow the creation of a mobile app that will login in a user and show the list of documents, give the ability to share them and add new documents from the mobile?
Does Sharepoint have such a API model?
To my knowledge, there is no such thing as "Document Management Programming Model". Document libraries, team sites, documents and workflows are all bits and pieces that belong to the SharePoint ecosystem.
What you have is object models that allow you to authenticate and retrieve contents, one of them being the REST API.
You can create your own app, but you might be interested in some alternatives, namely things that already exist and are out there ready to use (or get ideas from).
E.g.
Harmon.ie for Windows Phone (also available for iOS, Outlook, etc.)
