Is it possible to connect to a PowerPivot model in an Excel .xlsx file? (Not hosted on a SharePoint site... just the local file).
It must be, since Tableau can do it
Better yet, take a look at the C# source code of the Excel Refresh Service on codeplex - which demonstrates how to open and manipulate PowerPivot cubes that are embedded in Excel workbooks.
Be aware of what you're getting into... Excel automation has many quirks and tends to be unstable when used as a conduit into a PowerPivot cube.