'Template' is referring to the .xsn file created by InfoPath. The form file is usually referring to the .xml data file which is linked to the template.
When you publish a form to sharepoint, you are creating a document library. Under the document library will be a Forms/template.xsn file, which is the form template. This form template is automatically used when you click the 'Fill out this form' button or open an form already saved using this template.
I would recommend using the 'Publish' command to sharepoint, then test the form.
Also, to be have domain level trust, the sharepoint site must be recognized as in the domain. In IE, if the sharepoint site is in the 'Trusted' zone, it will not get domain trust. Domain trust requires that it be in the 'LocalIntranet' zone. If you reference the server as
http://foo, then it should by default be in the local intranet zone. If you reference it by
http://foo.com, it will default to internet zone.
Jim Cantwell
InfoPathDev - Redmond