1. You said you used an "update icon" to do your upgrade ... That's cardinal rule number 1 broken. DO NOT EVER USE ANY ONE-CLICK SITE-UPGRADE BUTTONS. Unless you did that with full knowledge of what it was going to do to your site, that was probably a big mistake.
Nevertheless, THAT issue has nothing to do with security concerns.

2. Using hosting-company-provided installations leaves *you* completely in the dark about how your site or your software works ... meaning you'll end up running scared anytime even the slightest thing doesn't work as expected.
It also leaves you in a position of "thus we do not have the original files". In fact, you can download the Zen Cart files from the Zen Cart website. Click the Home link to find the download link on the home page. There's also an FAQ article explaining where to download the code ... which can be found by searching "where can I download" in the FAQ area: https://www.zen-cart.com/tutorials/index.php?article=323

3. I don't know what the "BANNED_USERS" or "CACHE" or "CACHED_PERMISSIONS" tables are for. Zen Cart doesn't have tables named like that. So, I can't vouch for any concerns you may have about those messages. Anything related to those has nothing to do with Zen Cart.

Everything you're talking about seems more related to upgrade and software issues ... and has nothing to do with people accessing your site from abroad.

Documentation references on the process of correctly upgrading your Zen Cart software can be found here: https://www.zen-cart.com/tutorials/index.php?article=98