How can a new document tray be created automatically for users, which have been created via .NET API?
The creation of a default document tray for new users depends on the default organization role. To enable the creation assign a file cabinet to the default organization role and save the configuration. Afterwards document trays are automatically created for new users, who are created via .NET API. Following method shows how to create a new user in .NET API:

void CreateNewUser(Organization org, string userName, string email, string password)
    NewUser newUser = new NewUser
        DbName = userName,
        Name = userName,
        Email = email,
        NetworkId = "",
        Password = password