Hi -
Yes, your admin approved template needs to be re-uploaded to Central Admin - in this scenario, Quick Publish updates the network published template - when you publish for admin approval, you enter the network location the admin (maybe you!) will upload the published template from.... so all Quick Publish is doing is updating this file. Does that make sense?
I do not know what this means:
When I click on the form template in the Library it tries to open the form with InfoPath
This makes it sound as if you have saved the form template as a file in your SharePoint library as well as setting the library to use the content type that you admin approved? The correct way to open a form in SharePoint is Documents > New Document or the Add Document button in the library. The template is not saved as a file in the library. The XML created when a user fills out the template, on the other hand, should be saved in the library.