If your problem is that you're constantly being disconnected and reconnected, perhaps the fault is in your ISP or in the FTP client you're using. Are you on dial-up or broadband, what is your operating system, and what FTP client are you using? Even when I was on slow dial-up, I never had a problem with FTP Commander dropping me. If you're losing your connection just long enough to drop your FTP session, you should look for an FTP client that will resume an interrupted transfer rather than starting all over. Lots of people here swear by Filezilla, but I don't know if it has that feature.

If you're being blocked from the server, I would guess that your FTP client is opening multiple connections. Too many, and LP's security system thinks you're mounting a Denial of Service attack and blocks the IP address of your PC or ISP. Maybe the client is failing to close disconnected sessions -- another reason to try another FTP client.

While there is an option in cPanel > File Manager to extract (unzip) a zip file, I don't see anything to pack (zip) a directory.

If you don't mind working with .tgz files (gzipped tar files), you could probably write a shell script (or directly run a tar command) in cron to tar and zip selected directories or files. I think it's the -z flag in tar. I think there are some 'tar' utilities that run under Windows. Try doing an advanced  search on 'tar zip backup' or similar terms and see what comes up.