First, thanks for buying fH. It is much appreciated!
You are correct that fH does both google sync and supports several types of feed authentication, so don't worry what you are asking it SHOULD do
When does fH sync with google?
When you read an article (or change the state of the favorites, etc) INSIDE fH, a notification message is sent to your google reader account immediately. Large bulk actions (such as marking an entire folder as read) will take a few seconds in the background to complete... you can keep an eye on the network indicator to see the traffic happening.
When you read an article (or change the state of the favorites, etc) OUTSIDE fH, fH will not know about these changes until the feed is refreshed (Note that if you are staring at your google reader page when you read an article in fH, you will have to click on the feed name on the left again to have it refresh the articles list).
When does fH not sync an article?
If you downloaded articles before you enabled Google Sync, then those articles will not sync in either direction. Don't worry, eventually they will scroll off the face of the earth in fH as new articles become available. This is just how Google works: It cannot identify articles unless they are downloaded from Google (fH is handed a special id for the article).
If you read or change the favorites, etc. when offline, those changes are not sync'ed.
If the 'Sync with google' setting for the feed is set to OFF, no articles will be synced. If you later enable syncing, any of the old articles downloaded prior to the enabling will not be sync'ed (same reason as when you have google sync globally disabled).
If the feed uses HTTPS, or authentication, google in general rejects it (most likely you do not have these feeds in your google account because of this).
You could be experiencing a bug as well. THere is an update still not out of Apple's review process yet that does fix some things with Google Reader, but I cant say there were issues where reading did not get reflected in the Google account (I just double-checked to be sure).
I have subscribed to this thread for instant replies..... please post back if you have a bug or questions, am here to help. An update is being worked on and a bug with Google Sync would be very important to fix