Answers to some of your questions about synchronizing contacts between your Samsung Galaxy S III 4G LTE and Gmail
How does my Samsung Galaxy S® III 4G LTE back up and restore contacts with Gmail™?
If you set up a Google™ account on your phone, your Gmail contacts are automatically imported to your phone. After you've associated your Google account with your phone, anytime you add, import, or transfer a contact, you can choose where to save the contact. New contacts don't synchronize with your Google account unless you select Google.
If you add contacts to your phone before setting up a Google account, your contacts don't synchronize with Gmail. They save as Phone Only contacts.