As said above, there are multiple phonebooks.
Open the People app.
Go to Menu/View and you will see the phonebooks available and the options to turn them on/off.
Any contact that is in the "Phone" phonebook, WILL NOT sync with anything else and will only be available on the phone.
When you create a new contact the top drop down box is for "Contact Type" (this would be less confusing if it said Phonebook or something here) and it is set to "Phone" be default. You must change this to "Google" if you want the new contact to sync to Google.
If you have already created a bunch of contacts in the Phone phonebook, the easiest way to move them to the Google phonebook so that they sync is as follows.
Menu/Import Export / Export to SD / Phone
Now that the Phone contacts have been exported to SD, DELETE them from the phonebook.
Now Menu/Import Export / Import from SD / Google.
The contacts should now appear in the Google phonebook and sync with Google (it may take awhile or you can force a sync by going through Settings/Accounts).
As far as the facebook birthdays/phone numbers not transferring, the problem is with Facebook. FB will not allow you to export contact data from your account. So, the phone has permission to view the data, but not to copy and sync it. A while back someone wrote a program to rip the contact info from his facebook account and sync it with Plaxo, and FB shutdown his account within 24 hours (if you google i am sure you can find the blog about it). The ONLY way to get data from facebook to google is to manually enter it (which sucks).
*Something to note is that the Calendar works in a similar way. The default calendar is called "My Calendar" and is ONLY available on the phone. If you want an item to sync, you must add it to one of the google calendars.