I am wondering if Samba plays any role in this - otherwise, how would my Windows FTP client be able to communicate with a Linux server?
Windows Explorer simply has a built-in FTP ("File Transfer Protocol" - a protocol that handles file transfers all by itself) client - that is how it's communicating. Samba is not involved if the connection is explicitly through FTP.
For instance if I wanted to unzip a ZIP file on the server, would I be able to just right-click it as normal, or would I need a Linux equivalent to ZIP/TAR installed on the machine in order for the Unzip command to work?
If you right-click and unzip, your Windows machine would read the contents by downloading the file over FTP, then unzip it to the FTP site, so if you had write permissions: yes, you could unzip things this way.
It will, however, involve a file transfer to the Windows machine of the compressed data, and then an upload to the Linux machine of the uncompressed data, so you'd be much better off simply using PuTTY as an SSH client and running
unzip directly on the Linux machine.
Disclaimer: I don't even know if the built-in FTP client supports uploading over FTP. Don't think I've ever used that functionality, and haven't been on Windows for a long time, but a quick net search seems to indicate that it does indeed support uploading.