I'm sorry, but that error doesn't make sense to me. Mysqli_connect doesn't involve a file of Zen Cart, meaning it doesn't try to access a Zen Cart file so don't know how or why it is reporting: no such file or dir...
That said, did a search for "mysqli_connect HY000/2002 no such file or dir" and came up with several results associated with an inability to connect with the database server. The solutions seen include ensuring a correct reference to the database server, the correct port is used/identified, that the service is running or been restarted, etc...
Is this the error/response that occurs every time the shopping cart page is reloaded after making a country selection on the page and updating?
As to potential other issues, as said there were some other corrections applied after ZC 1.5.6a was released that affect this method of evaluating shipping options. The results of those (and other modifications) can be seen in the current github version of ZC 1.5.6: https://github.com/zencart/zencart/tree/v156
As I recall for a customer not logged in, files modified for more strict php requirements included:
includes/modules/shipping_estimator.php
includes/modules/pages/checkout_shipping/header_php.php
includes/classes/order.php
These and the other files modified seem to be covered by the following Pull Requests:
https://github.com/zencart/zencart/pull/1951/files
https://github.com/zencart/zencart/pull/1952/files
https://github.com/zencart/zencart/pull/2031/files
https://github.com/zencart/zencart/pull/2165/files
There are a few others regarding shipping modules and the absence of "proper" response, though not sure that those apply at the moment.
As to other sites with Zen Cart 1.5.6aish installation? They are out there and as I said the one(s) that I have do not currently have this issue, though they include the updated code as available on github and any other corrections that have been in-house identified but not yet shared with the github location.