Remove internal uses of fullPaths in the directory model in Files app.

This is the final CL removing the rest of fullPath dependencies in the directory model.

TEST=Tested manually. Hopefully all of the scenarios covered by this CL.
BUG=320967

Review URL: https://codereview.chromium.org/141433012

git-svn-id: svn://svn.chromium.org/chrome/trunk/src@246594 0039d316-1c4b-4281-b951-d872f2087c98
SVN changes up to revision 246594