I think nearly all of the log entries matching the examples shown in the document provided, relate to the parts of the skin template system you have stripped out (Language chooser, Featured box, Sale Items box, Newsletter box, Live Help, Debug, and Skin chooser).
Looking at the source HTML delivered to my browser, I see that there is a removed text entry field for the person's phone number.
This will cause the form to not even be submitted if the form validation fails.
In the file /skins/ho/js/3.cubecart.validate.js, at lines 66-97, there are the rules for the contact form.
Remove 92-94:
},
'contact[phone]': {
phone: $('#validate_phone').text()
Remove 83-85:
},
'contact[phone]': {
phone: true
What I do not know is this: How smart is the Validator? If the Validator does not find a text entry input field having the name contact['phone'], even with a rule referencing it, will the validator still submit the form?
According to this, the Validator does not check non-existing fields (from five years ago):
https://forum.jquery.com/topic/jquery-validate-check-that-a-field-exists-before-validating
In other words, I do not know if this edit will fix things.
FYI: The core code in cubecart.class.php, says only that 'email', 'name', 'subject', 'enquiry' are the only required fields. So maybe an issue will get posted in the Github that compares the Validator rule for the phone being required, but the core code does not.