It seems like I do not have access to many of the features how can I
activate the additional options such as the synchronize tab.

synchronize tab is enabled in Business edition and higher.
Please see:
https://www.geniusconnec...connect-license-options/ 

I'm considering ordering GeniusConnect, but I'm unsure which license I would need. If I'm linking an SQL database to contacts in an Outlook Public Folder, must the software be installed on each computer for employees to see (not admin) the updated information? Only one computer needs to be able to admin the processes, but all 12 of our employees need to be able to see the updated info in the public folder. Any suggestions?

Hello, you will need MultiUser license,you need only one installed computer,
since MS Exchange is distributing the data to all users.

If you need to sync. private non exchange folders (.pst) (not public folder), GeniusConnect has to be installed on every computer.