I have just made an iOS app that can share to Facebook. The Facebook post correctly shows that my sharing is coming from my app and shows \"1 hours ago via MYAPP\". The name of
In "Native iOS App" on your Facebook app settings just enable "Deep Linking". This would make facebook check if the app exists (probably by checking "canOpenURL" with your app URLScheme) if it exists it would open your app and if it doesn't it would send your users to iTunes.
Pay attention it could take a while till this change would be propagated.
I am not sure the link will bring you to the iTunes download page. According to my experience, that link will take you to the app's profile page on Facebook.
Something was changed since 1st Feb, 2012 that Facebook will not create profile page for new app by default. Please refer to this article - http://developers.facebook.com/blog/post/611/
However, you can manually create the profile page by going to "App Details > Contact Info", and then click the "Create Facebook Page" button as described in the article above.