I am using Google Analytics to track conversions of our 3rd-party hosted ecommerce site. I found that ALL the conversions were attributed to the website URL source and the referral medium even some of them were actually from PPC (AdWords campaign). Was that because the "thank-you" conversion page where the code was pasted belonged to the 3rd-party host site as opposed to our own domain? How should I fix it?
most of these solutions have a configuration that captures the information and passes it back with the confirmation information - and onwards through to Analytics.
If the gclid is not passed across then the shopping cart site only knows where the visitor came from - hence the referral attribution from your site.
Start with your shopping cart solution - look on their support pages - or send them the question - sometimes the solution is simple - other times it can be very complicated - but it can (almost) always be done.
It's to do with the cross domain tracking - they will be able to help you.
Since you are using Google Analytics to track conversions, you must also use the cross-domain tracking code. Basically, what this does is attach the gclid value to the form being submitted to the 3rd party. Analytics code on that site turns that into a cookie for that domain, and that allows the conversion tracking to happen properly. You can also set up the tracking so that both sites are tracked together on your domain Analytics.
You have 2 options:
Easiest option: Use the AdWords conversoin tracking code on the thank you page.
Still pretty easy option: Implement cross-domain tracking on both sites.