Hello KevinV_630,
Thank you for posting in Microsoft Community forum.
1.Based on the description "\Server1\Public$ - This server is running 2008 and this is the one I need to replace. I am also not sure why the folder target path has a "$" after it.", please check have you set this public folder as hidden folder.
2.Based on the description "If I make a new folder on a sub folder within "Public", from the namespace path (\Companyname.com\shared\Public), it only replicates to \Server3\Public and not the other servers. If I make a folder from \Server3\Public, it shows up on the namespace (\Companyname.com\shared\Public) but not the other servers.", it seems \Companyname.com\shared\Public syncs with \Server3\Public, and \Server3\Public syncs with \Companyname.com\shared\Public.
3.Based on the description "If I make a new folder on \Server2\Public or \Server1\Public, it never shows up on the namespace, \Server3\Public, etc. The just live on the server where they were placed.", it seems the the namespace and \Server3\Public is in sync, but the namespace and \Server3\Public do not sync with \Server2\Public or \Server1\Public (as mentioned in point 2).
I hope the information above is helpful.
If you have any question or concern, please feel free to let us know.
Best Regards,
Daisy Zhou